LoRA Probe Activations - Gate, Up, and Down Projections

All projections show: input activations ⋅ A matrix (rank-1 LoRA neuron activations)

Layer 0

GATE_PROJ

Top 16 Positive Activations
Rollout 43, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠC-NOTĠgate
Rollout 5, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊOneĠbaseĠofĠaĠtrapez
Rollout 63, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$P$ĠĠdenote
Rollout 94, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊDetermineĠwhichĠsetĠofĠstatesĠmentioned
Rollout 34, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊFindĠcontinuousĠfunctionsĠĠ$x(t
Rollout 60, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequency
Rollout 55, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĊ\[\mathbf{
Rollout 88, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊConsiderĠConvolutionalĠNeuralĠNetworkĠD
Rollout 84, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊGaryĠpurchasedĠaĠlargeĠbeverage,Ġbut
Rollout 47, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠtrainĠwithĠcross-sectionalĠareaĠ$
Rollout 78, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊJonĠandĠSteveĠrideĠtheirĠbicyclesĠalong
Rollout 11, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊSupposeĠĠ$a,\,
Rollout 12, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊProveĠthatĠtheĠfunctionĠ\[
Rollout 15, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$n\geq
Rollout 13, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊJoanieĠtakesĠaĠ$\$6
Rollout 14, Act: 0.539
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠfunctionĠĠ$f:[0
Top 16 Negative Activations
Rollout 86, Act: -1.989
1.04.ĊĊSoĠcase4:ĠâīĪ1.04.ĠWhileĠcase3ĠisâīĪ
Rollout 72, Act: -1.990
Ġ(sqrt(5)Ġ+1)/4ĠâīĪĠ(2.236Ġ+1)/4
Rollout 40, Act: -1.993
1421356Ġ+5ĠâīĪ19.1421356
Rollout 93, Act: -1.994
2=ÏĢ2Ġ/1007ĠâīĪĠ0.00987,Ġso
Rollout 4, Act: -1.996
.003606)^2ĠâīĪĠ(0.0036)^2Ġ=
Rollout 53, Act: -1.999
Ġ)=Ġ-243/73ĠâīĪĠ-3.333ĊĊThenĠx5
Rollout 72, Act: -2.000
Ġ-(sqrt(5)Ġ+1)/4ĠâīĪĠ-0.8090.ĊĊTherefore,
Rollout 22, Act: -2.000
Ġsee:Ġsqrt(8/(3ÏĢ))ĠâīĪĠsqrt(8/9.4248
Rollout 53, Act: -2.001
Ġcompute:ĊĊ-243/73ĠâīĪ-3.328,ĠandĠ-1
Rollout 53, Act: -2.002
=Ġ(-1557)/73ĠâīĪ-21.328ĊĊThereforeĠarea
Rollout 86, Act: -2.008
1/6)*(pi/2)^3ĠâīĪĠ(1/6)*(3.875
Rollout 69, Act: -2.016
1Ġ+Ġterm2Ġ+Ġterm3Ġ+...ĠâīĪ0.0348Ġ+0.0
Rollout 17, Act: -2.017
ĠâīĪ8ÏĢ*1.41ĠâīĪ11.28*3.14
Rollout 86, Act: -2.017
Ġ(pi/2)^3Ġ/3Ġ)ĠâīĪ(1/2)*(3.875
Rollout 17, Act: -2.022
.41ĠÃĹ10^-33ĠâīĪ8ÏĢ*1.41ĠâīĪ1
Rollout 69, Act: -2.056
3)/1Ġ=Ġ2Ġ+âĪļ3ĠâīĪ3.732,ĠwhichĠisĠnotĠclose

UP_PROJ

Top 16 Positive Activations
Rollout 47, Act: 1.331
ĠmassĠflowĠrateĠmustĠbeĠconserved,ĠsoĠÏģĠ*ĠA1Ġ*Ġv1Ġ=ĠÏģĠ*
Rollout 46, Act: 1.301
1ĠandĠM2,ĠeachĠatĠaĠdistanceĠÏģ1ĠandĠÏģ2ĠfromĠtheĠpointĠ(x
Rollout 25, Act: 1.297
ĠweĠassumeĠtheĠasteroidĠisĠaĠsphereĠwithĠdensityĠÏģ,ĠthenĠmassĠmĠ=Ġ(4/3)
Rollout 47, Act: 1.293
:ĊĊp0Ġ+Ġ(1/2)ĠÏģĠvt2Ġ=ĠpĠ+Ġ(1/2)
Rollout 47, Act: 1.281
2Ġ=ĠpĠ+Ġ(1/2)ĠÏģĠv_air2ĊĊThus,ĠtheĠpressureĠdifferenceĠp
Rollout 87, Act: 1.281
ĠaĠcornerĠofĠaĠcubeĠwithĠuniformĠchargeĠdensityĠÏģĠis:ĊĊVĠ=Ġ(ÏģĠa2)/(4
Rollout 47, Act: 1.280
0Ġ-ĠpĠ=Ġ(1/2)ĠÏģĠ[(Ġ(4/3ĠvtĠ)2Ġ-Ġvt
Rollout 47, Act: 1.278
Ġ-ĠpĠ=Ġ(7/(2N))ĠÏģĠvt2,ĠandĠweĠneedĠtoĠfindĠN.
Rollout 46, Act: 1.273
ĠthatĠUĠisĠgivenĠasĠsuch,ĠsoĠperhapsĠÏģ1ĠandĠÏģ2ĠareĠradialĠdistancesĠfromĠtwo
Rollout 47, Act: 1.271
0Ġ-ĠpĠ=Ġ(1/2)ĠÏģĠ(v_air2Ġ-Ġvt2)ĊĊWeĠalready
Rollout 47, Act: 1.262
2Ġ)Ġ]Ġ=Ġ(1/2)ĠÏģĠ(7/9Ġvt2Ġ)Ġ=Ġ(7
Rollout 46, Act: 1.261
ĠproblemĠhasn'tĠspecified,ĠmaybeĠweĠcanĠdefineĠÏģ1ĠandĠÏģ2ĠasĠfunctionsĠofĠxĠand
Rollout 46, Act: 1.258
ĤÏģ2/âĪĤx.ĠButĠIfĠÏģ1ĠisĠtheĠdistanceĠfromĠM1ĠtoĠtheĠpoint
Rollout 47, Act: 1.257
2Ġ)Ġ=Ġ(7/18)ĠÏģĠvt2ĊĊNow,ĠcomparingĠthisĠtoĠtheĠgiven
Rollout 47, Act: 1.256
2Ġ)Ġ]Ġ=Ġ(1/2)ĠÏģĠ[Ġ(16/9Ġvt2Ġ-Ġ
Rollout 46, Act: 1.255
ĠM1ĠandĠM2ĠareĠstationary.ĠThenĠÏģ1ĠandĠÏģ2ĠareĠtheĠdistancesĠfromĠthe
Top 16 Negative Activations
Rollout 52, Act: -0.456
<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠaboutĠyummyĠintegers.ĠTheĠquestionĠis:
Rollout 26, Act: -0.465
<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthisĠproblemĠout.ĠSo,ĠweĠhave
Rollout 18, Act: -0.468
<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblemĠstepĠbyĠstep.ĠFirst,ĠletĠme
Rollout 79, Act: -0.470
<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠcrosswordĠclue.ĠTheĠclueĠisĠ"Noticed
Rollout 91, Act: -0.474
<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠgeometryĠproblemĠstepĠbyĠstep.ĠTheĠquestionĠis
Rollout 45, Act: -0.480
)=8Ġ+16Ġ-4=20ĠâľĶï¸ıĊĊf(-4)=Ġ-64Ġ+
Rollout 45, Act: -0.488
64Ġ+64Ġ-4=Ġ-4ĠâľĶï¸ı.ĠOkay,Ġmatches.ĊĊSoĠf(x)
Rollout 45, Act: -0.493
(1)=1Ġ+4Ġ-4=1ĠâľĶï¸ıĊĊf(2)=8Ġ+16
Rollout 45, Act: -0.496
8+16-4=20,ĠâľĶï¸ı,Ġf(-4)=-64Ġ+
Rollout 45, Act: -0.499
16=55+4=59ĠâľĶï¸ı.ĊĊAnotherĠtest:Ġx=-1,Ġy
Rollout 45, Act: -0.499
(1)=1+4-4=1ĠâľĶï¸ı,Ġf(2)=8+16
Rollout 45, Act: -0.507
Ġ-64Ġ+64Ġ-4=-4ĠâľĶï¸ı.ĠAlsoĠf(0)=0Ġ+0
Rollout 45, Act: -0.519
(0)=0Ġ+0Ġ-4=-4ĠâľĶï¸ı.ĠSoĠworks.ĊĊButĠtoĠmakeĠsureĠthat
Rollout 45, Act: -0.522
Ġ+12)=Ġ-2Ġ+6=4ĠâľĶï¸ı.ĊĊYes,Ġmatches.ĠSoĠfunctionĠseemsĠokay
Rollout 23, Act: -0.528
<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠstepĠbyĠstep.ĠSo,ĠweĠhave
Rollout 54, Act: -0.541
<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestionĠstepĠbyĠstep.ĠFirst,ĠIĠneed

DOWN_PROJ

Top 16 Positive Activations
Rollout 49, Act: 1.277
ProveĠthatĠifĠeveryĠsubspaceĠofĠaĠHausdorffĠspaceĠĠ$X$ĠĠisĠ
Rollout 49, Act: 1.258
'sĠunpackĠtheĠtermsĠhere.ĊĊFirst,ĠaĠHausdorffĠspaceĠisĠaĠtopologicalĠspaceĠwhereĠany
Rollout 49, Act: 1.257
ĠstatementĠis:ĠIfĠeveryĠsubspaceĠofĠaĠHausdorffĠspaceĠXĠisĠÏĥ-compact,Ġthen
Rollout 49, Act: 1.251
.ĠTherefore,ĠifĠweĠcanĠshowĠthatĠaĠHausdorffĠspaceĠwhereĠeveryĠsubsetĠisĠFÏĥĠmust
Rollout 49, Act: 1.236
ĠisĠclosed.ĠButĠaĠperfectĠsetĠinĠaĠHausdorffĠspaceĠisĠuncountable.ĠMaybeĠthat
Rollout 49, Act: 1.232
ĠofĠclosedĠsets.ĠButĠwait,ĠinĠaĠHausdorffĠspace,ĠaĠÏĥ-compactĠsubspace
Rollout 57, Act: 1.224
Ġconcepts.ĠItsĠpropertiesĠincludeĠbeingĠcompletelyĠregular,ĠHausdorff,ĠbutĠnotĠnormal.ĠĊĊSinceĠthe
Rollout 49, Act: 1.221
Ġinstance,ĠinĠtheĠrealĠlineĠ(whichĠisĠHausdorff,Ġuncountable),ĠthereĠareĠsubsets
Rollout 49, Act: 1.220
Ġhere.ĊĊWait,ĠbutĠourĠspaceĠXĠisĠHausdorff,ĠnotĠnecessarilyĠmetrizable.ĠHmm
Rollout 49, Act: 1.220
Ġbe:ĠIfĠXĠisĠuncountableĠandĠHausdorff,ĠthenĠthereĠisĠaĠsubspaceĠthat
Rollout 49, Act: 1.219
ĠsubsetĠisĠÏĥ-compact,ĠthenĠinĠaĠHausdorffĠspace,ĠthatĠimpliesĠthatĠeveryĠsubsetĠis
Rollout 49, Act: 1.219
ĠÏĥ-compactĠbyĠhypothesis.ĠButĠinĠaĠHausdorffĠspace,ĠperfectĠsetsĠareĠclosed,Ġhence
Rollout 49, Act: 1.218
InĠsuchĠaĠspace,ĠconsiderĠX,ĠaĠHausdorffĠspaceĠwhereĠeveryĠsubsetĠisĠFÏĥ.
Rollout 49, Act: 1.218
,ĠaĠcommonĠexampleĠofĠanĠuncountableĠHausdorffĠspaceĠthatĠisn'tĠÏĥ-compactĠis
Rollout 49, Act: 1.218
ĠXĠisĠcountable.ĊĊWait,ĠinĠaĠHausdorffĠspaceĠwhereĠsingletonsĠareĠGδ,
Rollout 49, Act: 1.215
ĠforĠaĠtheorem.ĠIĠrecallĠthatĠifĠaĠHausdorffĠspaceĠisĠhereditarilyĠLindelö
Top 16 Negative Activations
Rollout 44, Act: -0.914
ĠthatĠcase,ĠirreversibleĠworkĠisĠhigher.ĊĊButĠinĠanotherĠexampleĠwhereĠinitialĠp2=1.5Ġatm
Rollout 40, Act: -0.915
=144,ĠremainderĠ6.ĠSoĠÏī^150=Ïī^(8*
Rollout 1, Act: -0.917
Ġx'Ġ-Ġy⣩Ġ=Ġ0.ĊĊSoĠanotherĠwayĠtoĠthinkĠofĠthisĠisĠthatĠtheĠsetĠ{
Rollout 21, Act: -0.919
ĠthoughtĠbefore,ĠwithĠonlyĠfiniteĠexpressionsĠallowed.ĠSoĠanotherĠapproach:ĠcanĠweĠcreateĠanĠequationĠthatĠmustĠhold
Rollout 62, Act: -0.920
ĠanotherĠnumberĠhasĠthreeĠprimeĠfactors.ĠSoĠevenĠifĠanotherĠnumberĠisĠsemiprimeĠ(twoĠprimes)
Rollout 57, Act: -0.921
ĠtoĠanalyzeĠtheĠtopologyĠgeneratedĠbyĠLebesgue-measurableĠsetsĠcontainingĠaĠpointĠ\(ĠxĠ\)ĠwithĠdensity
Rollout 41, Act: -0.921
ĠmyĠpreviousĠreasoning.ĠLet'sĠsee.ĊĊButĠperhapsĠanotherĠwayĠisĠusingĠcombinationsĠandĠpermutations.ĊĊHmm.ĠWait
Rollout 41, Act: -0.922
.ĠSoĠweĠneedĠtoĠhandleĠthisĠcarefully.ĊĊSoĠanotherĠwayĠis:ĊĊForĠthree-digitĠnumbersĠwithĠexactlyĠtwo
Rollout 57, Act: -0.923
ĠneighborhoodĠbasisĠconsistsĠofĠallĠsuchĠLebesgue-measurableĠsetsĠwithĠdensityĠ1ĠatĠx.ĠSo,
Rollout 93, Act: -0.930
ĠtoĠdifferentĠpowers.ĠHmm.ĊĊButĠmaybeĠthere'sĠanotherĠapproach.ĠForĠintegralsĠofĠcosineĠraisedĠtoĠhigh
Rollout 20, Act: -0.938
1)Ġâī¡1ĠmodĠm.ĊĊThenĠfollowingĠsimilarĠstepsĠasĠbefore:ĠmultiplyingĠbyĠ(p-1
Rollout 51, Act: -0.945
ĠisĠevenĠaroundĠpi,Ġlet'sĠsee.ĊĊButĠanotherĠview:ĠproductĠofĠsin8xĠandĠ|sin
Rollout 57, Act: -0.965
ĠtheĠfamilyĠofĠallĠdensityĠneighborhoodsĠatĠxĠ(measurableĠsetsĠwithĠdensityĠ1ĠatĠx)ĠformĠa
Rollout 8, Act: -0.969
Ġ10P5.ĠYeah.ĊĊButĠperhapsĠanotherĠperspective:ĠifĠshe'sĠselectingĠ5ĠcardsĠfrom
Rollout 57, Act: -0.974
ĠmeasureĠone.ĠSuchĠsetsĠareĠcalledĠsaturatedĠnonmeasurableĠsets,ĠbutĠareĠtypicallyĠonlyĠpossibleĠunderĠcertainĠset
Rollout 81, Act: -1.014
ĠatĠA,ĠedgesĠABĠandĠAC.ĠSotriangleĠareaĠcanĠalsoĠbeĠwrittenĠas:ĊĊĠArea=1

Layer 1

GATE_PROJ

Top 16 Positive Activations
Rollout 49, Act: 2.323
orff)ĠandĠitĠisĠaĠHereditarilyseparableĠHSĠspace,ĠbutĠIĠdon'tĠthinkĠwe
Rollout 33, Act: 2.270
.ĊĊk=3:ĠpermutationsĠwithĠfirstĠthreeĠpermutationof1-3:Ġ6*1=6
Rollout 54, Act: 2.248
ilesĠ(1.Ġ4-methylcyclohexan-1-olate,Ġ2.
Rollout 25, Act: 2.247
olaĠwithĠperigeeĠr_pĠandĠtheĠapseĠlineĠisĠwhereĠtheĠclosestĠapproachĠoccurs.ĠTheĠparameters
Rollout 42, Act: 2.236
ĠfinalĠproductĠproducedĠwhenĠcyclobutyl(cyclopropyl)methanolĠreactsĠwithĠphosphoricĠacid
Rollout 57, Act: 2.229
Wait,ĠperhapsĠaĠmethodĠusedĠinĠprovingĠUrysohn'sĠlemma.ĠButĠinĠaĠnormalĠspace,
Rollout 44, Act: 2.227
ĠreversibleĠprocess,ĠsinceĠtheĠexternalĠpressureĠisĠalwaysĠinfinitesimallyĠlargerĠthanĠtheĠinternalĠpressure.ĠIfĠthe
Rollout 42, Act: 2.178
ĠtheĠproductĠformedĠwhenĠcyclobutyl(cyclopropyl)methanolĠreactsĠwithĠphosphoricĠacid
Rollout 24, Act: 2.167
.Ċ<|im_start|>userĊConsiderĠanyĠrectangularĠtableĠhavingĠfinitelyĠmanyĠrowsĠandĠcolumns,ĠwithĠaĠrealĠĊĊ
Rollout 10, Act: 2.154
ĠyouĠaddĠtoĠS,ĠyouĠmustĠexcludeĠonlyĠfinitelyĠmanyĠothersĠ(i.e.,ĠnĠ+s
Rollout 46, Act: 2.154
ĠasĠ(acceleration_gravityĠ+Ġacceleration_centrifugal).ĠBecause:ĊĊForĠeachĠterm:ĊĊâĪĩ(-
Rollout 57, Act: 2.149
Ġlemma.ĠButĠinĠaĠnormalĠspace,ĠUrysohn'sĠlemmaĠholds.ĠHowever,ĠthisĠspaceĠisn
Rollout 65, Act: 2.146
Ġsolution.ĊĊĠButĠholdĠon,ĠsolutionĠfromĠmathĠconnoisseursĠshowsĠthatĠacceptableĠsimilarĠprismsĠrequireĠharmonic
Rollout 87, Act: 2.143
ĠafterĠallĠthis,ĠifĠIĠcannotĠproceedĠfurtherĠanalytically,ĠandĠtheĠproblemĠlikelyĠexpectsĠusingĠtheĠintegral
Rollout 37, Act: 2.142
015!ĠForĠanyĠintegerĠsubstitution,ĠneedtoĠseeĠitsĠCoefficientsĠareĠdivisibleĠbyĠ2
Rollout 49, Act: 2.141
ĠuncountableĠandĠhereditariyĠLindelof,ĠthenĠitĠhasĠaĠuncountablesubsetĠwith
Top 16 Negative Activations
Rollout 86, Act: -1.587
ĠisĠ1Ġ-0=1.ĊĊThus,ĠintegralĠofĠxĠsinĠxĠdxĠisĠ1.ĠTherefore
Rollout 51, Act: -1.587
3Ġ=Ġ-4/63ĊĊTherefore,ĠIntegralĠBĠ=Ġ-4/63ĊĊThus,
Rollout 86, Act: -1.587
1/2)cos2x.ĊĊThus,Ġintegral=Ġ-Ġ[Ġ-Ġ(1/2)xĠcos
Rollout 86, Act: -1.591
2Ġdx,Ġv=sinĠx.ĊĊThus,ĠintegralĠ=Ġx3ĠsinĠxĠ|ĠfromĠ0Ġto
Rollout 51, Act: -1.598
,ĠleadingĠtoĠ-4/63.ĊĊInĠIntegralĠBĠsteps:ĊĊĠsin9xĠ-Ġsin7x
Rollout 86, Act: -1.600
1/2)Ġsin2x.ĊĊThus,ĠintegralĠbecomesĠ(x2)(1/2)Ġsin
Rollout 86, Act: -1.602
1/2)Ġsin2x.ĊĊThus,Ġintegral=Ġx2*(1/2Ġsin2x
Rollout 86, Act: -1.616
5.ĊĊThus,ĠplugĠbackĠintoĠcase6:ĊĊĠintegralĠofĠx^2Ġcos^2xĠdx=(
Rollout 51, Act: -1.650
ÏĢ}Ġcos8tĠsintĠdtĊĊTherefore,ĠIntegralĠBĠbecomes:ĊĊâĪ«0^ÏĢĠcos8x
Rollout 51, Act: -1.661
Ġsin7x]Ġ/Ġ2ĊĊTherefore,ĠIntegralĠBĠoverĠ0ĠtoĠÏĢĠis:ĊĊ2Ġ*
Rollout 51, Act: -1.664
Ġcos9x]Ġ/Ġ2ĊĊTherefore,ĠIntegralĠAĠoverĠ0ĠtoĠÏĢĠisĠâĪ«0
Rollout 51, Act: -1.670
|ĠdxĠ=Ġ-4/63ĊĊSoĠIntegralĠBĠ=Ġ-4/63.ĊĊSoĠyes
Rollout 51, Act: -1.672
sinx|Ġ=Ġ-sinx.ĊĊTherefore,ĠIntegralĠAĠcanĠbeĠwrittenĠas:ĊĊâĪ«0^ÏĢ
Rollout 87, Act: -1.676
Ġ/Ġsqrt(x2Ġ+Ġy2)).ĊĊThenĠIntegralĠbecomes:ĊĊVĠ=Ġ(Ïģ/(4ĠÏĢĠε
Rollout 51, Act: -1.687
8xĠ|sinx|ĠdxĊĊSimilarly,ĠIntegralĠB:ĠâĪ«02ÏĢĠcos8x
Rollout 51, Act: -1.701
Ġ-cos(nx)/nĠ+ĠCĊĊSo,ĠIntegralĠBĠbecomes:ĊĊ[-cos9x/9Ġ+

UP_PROJ

Top 16 Positive Activations
Rollout 25, Act: 1.391
*(6.67e-11Ġ*ĠmĠ)/v2ĊĊSolvingĠforĠm,ĊĊ
Rollout 25, Act: 1.368
=6.849e14âĨĴĠM_eĠ+m=6.849e
Rollout 22, Act: 1.349
(8Ġ*ĠRĠ*ĠTĠ/Ġ(ÏĢĠ*ĠM))Ġ=Ġsqrt(8Ġ*Ġ8.3
Rollout 27, Act: 1.348
6ĠkgĠ*Ġ(1.49e5Ġm/s)^2)Ġ/Ġ(3Ġ*Ġ1
Rollout 22, Act: 1.327
(8Ġ*ĠRĠ*ĠTĠ/Ġ(ÏĢĠ*ĠM)).ĊĊRĠ=Ġ8.314
Rollout 27, Act: 1.327
timesĠ10^{5}Ġ\text{Ġm/s}}$ĠandĠtheĠestimatedĠcriticalĠtemperatureĠ$T
Rollout 22, Act: 1.326
(8Ġ*ĠRĠ*ĠTĠ/Ġ(ÏĢĠ*ĠM)),ĠwhereĊR=8.314
Rollout 27, Act: 1.326
timesĠ10^{5}Ġ\text{Ġm/s}}\)ĠandĠtheĠestimatedĠcriticalĠtemperatureĠ\(ĠT
Rollout 25, Act: 1.323
ĠâīĪĠ1.094e6ĠmĊĊDenominator:ĊĊsqrt(1Ġ+Ġ(Ġ(
Rollout 3, Act: 1.307
ĠasĠ\boxed{169Ġ\text{Ġm/s}}.ĊĊHowever,ĠdependingĠonĠsourceĠofĠproblem
Rollout 27, Act: 1.307
ĪĠ1.4915e5Ġm/s.ĠSoĠapproxĠ1.49e
Rollout 27, Act: 1.304
ĠâīĪĠ1.491e5Ġm/s.ĊĊYes,ĠsoĠapproximatelyĠ1.4
Rollout 27, Act: 1.301
Ġ2.2201e10Ġm2/s2ĊĊMultiplyĠbyĠmass:ĊĊ1.
Rollout 66, Act: 1.296
,Ġm2Ġ-n2Ġ)Ġ=Ġgcd(Ġm2Ġ+n2,Ġ(m2Ġ+n
Rollout 66, Act: 1.289
m2Ġ-n2Ġ)Ġ)Ġ=Ġgcd(Ġm2Ġ+n2,Ġ2n2Ġ)
Rollout 25, Act: 1.288
2Ġ=Ġ86.49e6Ġm2/s2ĊĊNumerator:Ġ10
Top 16 Negative Activations
Rollout 70, Act: -2.066
0.ĠThenĠusingĠthis,Ġlet'sĠproceed.ĊĊThus:ĊĊEquation:314d3Ġ+
Rollout 83, Act: -2.070
ĠisĠaĠsymmetryĠthatĠcyclesĠtheĠthreeĠresidueĠgroups.ĊĊThus,ĊĊTheĠrotationalĠgroupĠisĠofĠtheĠnonagonĠacting
Rollout 65, Act: -2.071
ĠleadsĠtoĠcube.ĠSoĠonlyĠthisĠsolutionĠexists.ĊĊThus,ĠthisĠapproachĠalsoĠyieldsĠcube.ĠSoĠcuttingĠalong
Rollout 27, Act: -2.073
Ġdensity.ĠHowever,ĠproblemĠmentionsĠnoĠsuchĠnumber.ĊĊThusĠpossibly,ĠinĠabsenceĠofĠmoreĠdetails,ĠproceedingĠwith
Rollout 20, Act: -2.073
ĠforĠ2,ĠwhichĠappearsĠonceĠinĠboth.ĊĊThusĠLCMĠisĠ2*3*7*
Rollout 68, Act: -2.076
ĠisĠinĠtheĠdomain.ĠHence,ĠoneĠsolution.ĊĊThus,ĠinĠgeneral,ĠforĠanyĠk<0,
Rollout 35, Act: -2.078
418,ĠwhichĠisĠwhatĠwasĠrequired.ĊĊThus,ĠthisĠseemsĠlikeĠtheĠlogicalĠconclusion,ĠandĠthe
Rollout 1, Act: -2.084
ĠwithĠcountablyĠorĠuncountablyĠmanyĠvertices.ĊĊIndeed,ĠinĠanĠinfinite-dimensionalĠHilbertĠspace,Ġone
Rollout 96, Act: -2.086
ĠtheĠoriginalĠpositive/right/eastĠinĠoriginalĠcoordinates.ĊĊThus,ĠpointĠP(-14,27)
Rollout 67, Act: -2.089
ĠSoĠsequenceĠpeaksĠaroundĠthere,ĠandĠthenĠdecreases.ĊĊThus,ĠtheĠtermĠn=3ĠisĠ18
Rollout 83, Act: -2.091
ĠbyĠrotationĠnumber3,ĠwhichĠcyclesĠtheĠgroups.ĊĊThus,ĠpermutationĠofĠ(group0,Ġgroup1,
Rollout 57, Act: -2.092
ĠHowever,ĠthisĠisĠsimilarĠtoĠourĠpreviousĠarguments.ĊĊIndeed,ĠifĠtakeĠPĠ=Ġirrationals,Ġthen
Rollout 67, Act: -2.101
ĠtheĠeighthĠtermĠremainsĠconsistentĠwithĠtheĠquadraticĠmodel.ĊĊThus,ĠtheĠeighthĠtermĠofĠtheĠsequenceĠisĠ\(\
Rollout 12, Act: -2.102
ĠorĠincompleteĠellipticĠintegral.ĠLet'sĠcheck.ĊĊIndeed,ĠinĠtheĠintegral:ĊĊâĪ«_{ν}^
Rollout 21, Act: -2.140
ĠstatementĠrequiresĠequationsĠonlyĠ(E=0).ĠInequalitiesĠaren'tĠallowedĠasĠtheyĠdon'tĠfitĠinto
Rollout 73, Act: -2.162
.ĠSoĠthisĠverificationĠalsoĠgivesĠtheĠsameĠresult.ĊĊThus,ĠtheĠsumĠNĠisĠ1639

DOWN_PROJ

Top 16 Positive Activations
Rollout 79, Act: 0.666
'tĠmakeĠsense).ĠNo.ĊĊAlternatively,Ġ"OrganismĠHen".ĠNo,ĠnotĠmakingĠsense.ĊĊAlternatively,
Rollout 98, Act: 0.613
ĠmillionĠyears.ĠTheĠwhiteĠdwarfĠwouldĠbeĠtheĠremnantĠafterĠtheĠstarĠhasĠshedĠitsĠenvelope.ĠNow,
Rollout 98, Act: 0.600
ĠWait,Ġwait.ĠAĠwhiteĠdwarfĠisĠtheĠremnantĠofĠaĠstarĠthatĠhasĠshedĠitsĠouterĠlayers.
Rollout 98, Act: 0.600
ĠA0V.ĠTheĠwhiteĠdwarfĠisĠtheĠremnantĠofĠaĠB5VĠstar.ĠSinceĠmassiveĠstars
Rollout 98, Act: 0.595
ĠaĠWD.ĠSoĠifĠtheĠWDĠisĠtheĠremnantĠofĠaĠB5ĠmainĠsequenceĠstarĠ(massĠ~
Rollout 98, Act: 0.512
ĠWait,Ġactually,ĠtheĠinitialĠmassĠleadingĠtoĠWDĠremnantsĠisĠusuallyĠroughlyĠupĠtoĠ8Msun.
Rollout 98, Act: 0.511
.ĠIfĠDA4ĠisĠaĠwhiteĠdwarfĠ(remnantĠofĠaĠstar)ĠandĠL4ĠisĠaĠbrown
Rollout 65, Act: 0.486
.ĊĊĠButĠholdĠon,ĠsolutionĠfromĠmathĠconnoisseursĠshowsĠthatĠacceptableĠsimilarĠprismsĠrequireĠharmonicĠmean
Rollout 57, Act: 0.471
PåĴĮPâĢĻæĺ¯å¼ĢçļĦãĢĤèĭ¥ä»ĸ们æĺ¯äºĴä¸įçĽ¸äº¤,èĥ½åIJ¦çĶ¨å¼ĢéĽĨå°ĨQ
Rollout 29, Act: 0.452
ĠtheĠbottomĠofĠtheĠjugĠ(mainĠpart)ĠshouldĠnormallyĠbeĠrho*g*h.ĠHowever,ĠifĠtheĠhandle
Rollout 29, Act: 0.446
ĠhĠisĠtheĠdepthĠbelowĠtheĠsurface.ĠSo,Ġnormally,ĠifĠyouĠhaveĠaĠcontainerĠopenĠtoĠtheĠatmosphere
Rollout 4, Act: 0.446
Ġ2.6%ĠtotalĠspread.ĠButĠsinceĠnormallyĠuncertaintyĠisĠdefinedĠasĠ±ĠhalfĠtheĠtotalĠspread,
Rollout 80, Act: 0.441
First,ĠtheĠproblemĠstatesĠthatĠneutrinos,ĠwhichĠnormallyĠrarelyĠinteract,ĠareĠnowĠbeingĠcompletelyĠabsorbedĠbyĠthe
Rollout 29, Act: 0.439
ĠaĠmanufacturingĠdefect".ĠSoĠtheĠhandle,ĠwhichĠisĠnormallyĠopenĠ(soĠthatĠwhenĠyouĠpour,ĠmilkĠcan
Rollout 99, Act: 0.427
573.ĊĊButĠsinceĠafterĠtwoĠiterations,ĠnormallyĠwouldĠtoĠcomputeĠx4,ĠasĠIĠearlierĠshowed
Rollout 15, Act: 0.426
ĠifĠexactlyĠoneĠneighborĠisĠ1Ġ(countedĠnormally),ĠbutĠconsiderĠtwoĠneighbors:Ġ"exactlyĠone
Top 16 Negative Activations
Rollout 97, Act: -0.547
ĠsinceĠwe'reĠdealingĠwithĠpressure,ĠmaybeĠusingĠpressureĠinsteadĠofĠconcentration.ĠSinceĠinĠaĠgasĠphaseĠreactionĠat
Rollout 30, Act: -0.550
Ġrelations,ĠthatĠmightĠbeĠchallenging.ĊĊWait,ĠperhapsĠinsteadĠofĠtryingĠtoĠfindĠexplicitĠexpressionsĠforĠa_nĠand
Rollout 22, Act: -0.551
Ġthere'sĠanotherĠformulaĠifĠweĠuseĠtheĠgasĠconstantĠinsteadĠofĠBoltzmann'sĠconstant.ĠLet'sĠcheck
Rollout 67, Act: -0.551
2Ġ+ĠBnĠ+ĠC.ĠThus,ĠmaybeĠinsteadĠofĠsolvingĠforĠtheĠvariablesĠa,Ġd,Ġp
Rollout 42, Act: -0.556
ocation.ĠLet'sĠconsiderĠthat.ĊĊAlternatively,ĠperhapsĠinsteadĠofĠaĠcarbocationĠrearrangement,ĠtheĠeliminationĠoccurs
Rollout 81, Act: -0.558
/7âīĪ3.14,ĠsoĠnegativeĠinstead.ĊĊThus,theĠambiguityĠarisesĠbecauseĠtheĠangleĠcomputedĠvia
Rollout 99, Act: -0.558
ĠquestionĠsaysĠ"decimalĠvalue",ĠdoesĠitĠwantĠdecimalĠinsteadĠofĠfraction?ĠButĠgivenĠtheĠquestionĠasksĠforĠdecimal
Rollout 28, Act: -0.560
ĠwhyĠallĠtheseĠconicsĠmustĠbeĠhyperbolasĠinsteadĠofĠotherĠtypesĠlikeĠellipsesĠorĠparab
Rollout 23, Act: -0.563
Ġcarefully.ĊĊButĠsinceĠthisĠstepĠisĠcritical,ĠmaybeĠinstead,ĠapproximateĠln(A).ĠForĠanswerĠD,Ġ
Rollout 23, Act: -0.564
.ĊĊAh!ĠIfĠsomeoneĠmistakenlyĠcomputedĠtheĠTotalĠactivityĠinsteadĠofĠjustĠalphaĠactivity,ĠtheyĠwouldĠgetĠTotalĠactivity
Rollout 16, Act: -0.570
ĠinĠthisĠcase,Ġit'sĠaĠbitĠdifferentĠbecauseĠinsteadĠofĠcollectingĠoneĠcouponĠatĠaĠtime,ĠeachĠsap
Rollout 11, Act: -0.585
ĠAltizio,ĠwhichĠsuggestsĠthereĠisĠaĠcleverĠsolutionĠinsteadĠofĠheavyĠcomputation.ĊĊRecallĠthatĠfromĠabcĠ=
Rollout 86, Act: -0.588
ĠsinxĠandĠcosxĠmultiplicativeĠtermsĠasĠpossibleĠinsteadĠofĠx.ĊĊBecauseĠsinxĠandĠcosxĠare
Rollout 99, Act: -0.591
antĠmethodĠbutĠusesĠaĠquadraticĠinterpolationĠthroughĠthreeĠpointsĠinsteadĠofĠaĠlinearĠone.ĠThisĠallowsĠitĠtoĠpotentially
Rollout 74, Act: -0.600
."ĠMaybeĠtheĠmodelĠhereĠisĠconsideringĠpermanentĠmagnetismĠinsteadĠofĠinduced.ĠPermanentĠship'sĠmagnetismĠwouldĠproduce
Rollout 50, Act: -0.609
ĠsomeĠway.ĠMaybeĠweĠshouldĠfindĠaĠbetterĠapproachĠinsteadĠofĠexpandingĠallĠthis.ĊĊAlternatively,ĠperhapsĠstartĠover

Layer 2

GATE_PROJ

Top 16 Positive Activations
Rollout 37, Act: 1.251
[Ċ[a_1^{k_1}ĠĠa_1^{k_2}ĠĠ...Ġ
Rollout 99, Act: 1.208
c)/(bĠ±Ġsqrt(b2Ġ-Ġ4ĠaĠc))ĊĊSoĠthat'sĠtheĠformula.ĠLet's
Rollout 99, Act: 1.161
c)/(bĠ±Ġsqrt(b2Ġ-Ġ4ĠaĠc))ĊĊFirst,ĠcomputeĠdiscriminantĠDĠ=Ġb
Rollout 37, Act: 1.126
]Ċ[a_2^{k_1}ĠĠa_2^{k_2}ĠĠ...Ġ
Rollout 37, Act: 1.088
015}^{k_1}ĠĠa_{2015}^{k_2
Rollout 11, Act: 1.029
Ġ=Ġ|b|^2*(|b|^4ĠaĠ)Ġ)Ġ=Ġ|b|^6ĠaĠâĨĴĠTherefore
Rollout 40, Act: 1.020
_{k=0}^{n-1}Ġa^kĠ).ĠThenĠusingĠgeometricĠseriesĠformula:Ġ(
Rollout 30, Act: 1.016
1}]Ġ/Ġ[a_{n+1}Ġa_{n+3}].ĠThen,ĠtheĠterm
Rollout 17, Act: 1.016
{r})=\frac{e^{-rĠ/Ġa_{0}}}{\sqrt{\piĠa_{
Rollout 40, Act: 1.015
_{k=0}^{n-1}Ġa^k}.ĠThisĠisĠassumingĠaĠâīłĠ1
Rollout 12, Act: 1.011
Ġ1Ġ/Ġsqrt(Ġ(x^2Ġ-Ġa^2)(b^2Ġ-Ġx^2
Rollout 14, Act: 1.001
Ġ)Ġ]3Ġ+3Ġ[sqrt(2ĠKĠaĠ)]^2Ġ(a/2Ġ)Ġ+3
Rollout 30, Act: 0.998
âĪŀĠ1/(Ġ(a_mĠ-1Ġ)Ġa_mĠ).ĠButĠthat'sĠtheĠsameĠasĠtheĠoriginal
Rollout 14, Act: 0.995
/2Ġ)Ġ+3Ġ(sqrt(2ĠKĠaĠ))(a2/4Ġ)Ġ+a3
Rollout 30, Act: 0.986
Ġ=Ġ1/(Ġ(a_mĠ-1Ġ)Ġa_mĠ)ĊĊWhichĠsuggestsĠthatĠtheĠsumĠcanĠbeĠrewritten
Rollout 30, Act: 0.979
ĠfirstĠsumĠintoĠ1Ġ+ĠsumĠ[1/(Ġa_k(a_kĠ-1Ġ)Ġ]ĊĊTheĠsecondĠsum
Top 16 Negative Activations
Rollout 56, Act: -2.765
+1})âĪļ3.ĠSoĠasĠnĠapproachesĠinfinity,ĠS=âĪļ3.ĠTherefore,ĠtheĠsuprem
Rollout 81, Act: -2.767
ĠabsoluteĠvalueĠgives22/7.ĠInĠtheĠdenominatorĠcaseĠh2Ġ-51<0,Ġso
Rollout 99, Act: -2.771
ĪĠ2.657ĊĊSoĠtheĠdenominatorĠisĠbĠ±Ġsqrt(D).ĠWhichĠisĠ-0
Rollout 51, Act: -2.793
/9Ġ-2/7ĊĊConvertĠtoĠcommonĠdenominator:ĊĊ2*(7Ġ-9)/63Ġ=
Rollout 90, Act: -2.802
7-digitĠnumber,ĠwhichĠmustĠstartĠwithĠaĠnon-zeroĠdigit,ĠhenceĠNĠmustĠbeĠfromĠ10
Rollout 53, Act: -2.806
29/292:ĊĊConvertĠtoĠcommonĠdenominator:ĊĊ81/4=Ġ(81Ġ*
Rollout 93, Act: -2.818
)}{1007}}\)ĠisĠnon-zeroĠonlyĠwhenĠ\(1007Ġ-Ġj\)
Rollout 16, Act: -2.827
192/31:ĠconvertĠtoĠcommonĠdenominatorĠ7*31=217:
Rollout 16, Act: -2.829
ĠbeforeĠconcluding,ĠdoubleĠcheckingĠtheĠcalculationĠstepĠwhereĠcommonĠdenominatorĠwasĠ1953,Ġso:ĊĊ1
Rollout 16, Act: -2.856
SoĠwritingĠallĠinĠtermsĠofĠfractionsĠwithĠaĠcommonĠdenominatorĠ(thoughĠmaybeĠweĠcanĠaddĠthemĠstepĠbyĠstep
Rollout 81, Act: -2.861
>0,ĠsoĠangleĠisĠacute.ĊĊButĠinĠdenominatorĠ[1Ġ-ĠtanĠalphaĠtanĠbeta],ĠasĠper
Rollout 16, Act: -2.864
Ġ64/63ĊĊConvertĠtoĠcommonĠdenominator.ĠTheĠdenominatorsĠareĠ217Ġand
Rollout 16, Act: -2.880
Ġ-64/63:ĠconvertĠtoĠcommonĠdenominatorĠ217*63=13
Rollout 99, Act: -2.895
6431374ĊĊThenĠtheĠdenominatorĠis:ĊĊOption1:ĠbĠ+Ġsqrt(D)=
Rollout 16, Act: -2.919
Ġ-1/63.ĊĊThenĠconvertingĠtoĠcommonĠdenominatorĠ1953:ĊĊ11/2
Rollout 16, Act: -2.932
}Ċ\]ĊĊConvertingĠallĠtermsĠtoĠaĠcommonĠdenominatorĠandĠsimplifying,ĠweĠfindĠtheĠexpectedĠnumberĠof

UP_PROJ

Top 16 Positive Activations
Rollout 20, Act: 1.863
Ġm,ĠweĠneedĠeither:ĊĊ-ĠIfĠpĠâī¡1ĠmodĠm:ĠthenĠknĠâī¡0Ġmod
Rollout 93, Act: 1.848
1007.ĊĊThen:ĊĊIfĠmĠâī¡0Ġmod1007,ĠthenĠsum_{
Rollout 2, Act: 1.837
WeĠneedĠtoĠsolveĠforĠxĠin:ĊĊxĠâī¡Ġ109Ġmod125ĊĊx
Rollout 20, Act: 1.829
Ġâī¡0ĠmodĠmĊĊ-ĠIfĠpĠâī¡ĠotherĠresiduesĠmodĠm:Ġp^{kn}Ġâī
Rollout 20, Act: 1.828
43.ĊĊSubcases:ĊĊ-ĠIfĠpĠâī¡1Ġmod43:ĠsumĠisĠkn+1
Rollout 20, Act: 1.824
43.ĊĊFirst,ĠnoteĠthatĠifĠpĠâī¡Ġ0ĠmodĠ43,ĠthenĠpĠ=
Rollout 2, Act: 1.818
Ġâī¡5Ġmod8.ĊĊTherefore,ĠPĠâī¡Ġ5Ġmod8.ĊĊAlright,ĠnowĠontoĠmodulo
Rollout 10, Act: 1.783
3Ġmod4:ĊĊPossibleĠmoves:ĊĊIfĠnĠâī¡0Ġmod4:Ġsubtract0Ġor1.ĊĊĠsubtract
Rollout 10, Act: 1.783
;0Ġ-1=3;ĊĊIfĠnĠâī¡1Ġmod4:Ġsubtract1Ġor0.ĊĊn
Rollout 20, Act: 1.779
=knĠ+1.ĠTherefore,ĠifĠpĠâī¡1ĠmodĠm,ĠweĠneedĠknĠ+1Ġâī
Rollout 93, Act: 1.773
ĠsoĠsumĠisĠN.ĠThus:ĊĊIFĠmĠâī¡0ĠmodĠNĠ=>Ġsum_{k=1}
Rollout 20, Act: 1.769
Ġ+1ĠmodĠm.ĠSo,ĠifĠpĠâī¡1ĠmodĠm,ĠthenĠÏĥ(p^{kn})
Rollout 20, Act: 1.766
Ġ+Ġ...Ġ+Ġp^{kn}.ĠIfĠpĠâī¡1ĠmodĠmĠ(meaningĠmĠisĠ43
Rollout 90, Act: 1.759
Ġinclusive,ĠweĠcanĠexpressĠthisĠas:ĊĊMĠâī¡Ġ-9NĠmodĠ1009ĊĊ
Rollout 2, Act: 1.758
ĠPĠmodĠ8.ĊĊSinceĠ10Ġâī¡Ġ2ĠmodĠ8,ĠsoĠ10
Rollout 70, Act: 1.753
.ĊĊWait,ĠisĠthatĠcorrect?ĊĊBecauseĠNĠâī¡d0Ġmod7Ġ(sinceĠN=Ġ...Ġ+
Top 16 Negative Activations
Rollout 42, Act: -1.647
ĠthreeĠsubstituents:ĠoneĠhydroxylĠ(fromĠtheĠalcohol),ĠaĠcyclobutylĠring,
Rollout 26, Act: -1.651
fourĠsigĠfigs),ĠtheĠEnthalpyĠofĠformationĠofĠHg(g)ĠisĠgivenĠasĠ6
Rollout 98, Act: -1.655
.85Msun),ĠK3V,ĠAĠstarĠwithĠaĠmassĠofĠ0.9Ms
Rollout 79, Act: -1.658
ĠAĠcommonĠtermĠthatĠmeansĠinvestigators'ĠfacilityĠisĠ"OperationsĠCenter".ĠLet'sĠcheck.ĠOperationsĠ=Ġ1
Rollout 7, Act: -1.660
56*(N/3)-42*(Ġ(N/3)Ġ-x).ĊĊSimplify:ĊĊ
Rollout 56, Act: -1.664
esyuĠstartsĠwithĠaĠright-angledĠtriangleĠ\(ĠA_0A_1A_2Ġ\)
Rollout 25, Act: -1.664
Ġ=Ġ3175e3ĠmĠ(fromĠpreviousĠcalcs),ĠthenĠrĠ<Ġr_p_final
Rollout 25, Act: -1.667
Ġ=Ġ3175e3ĠmĠ(fromĠearlierĠcalculation)ĊĊr_p_finalĠ=Ġ63
Rollout 25, Act: -1.684
080,625e6Ġ(fromĠearlier,ĠwhenĠweĠcalculatedĠr_p_initial)Ċ(
Rollout 98, Act: -1.699
ĠwhenĠtheĠB5ĠstarĠbecomesĠaĠWD.ĠBinaryĠinteractionsĠorĠmassĠlossĠscenariosĠallowĠforĠplausibleĠcoexistence.ĊĊ
Rollout 98, Act: -1.701
ĠandĠ**M4V**Ġ(redĠdwarf,ĠPopulationĠI)ĠstarsĠareĠyoungerĠandĠmetal-rich.ĠThese
Rollout 25, Act: -1.727
1.008e13Ġ*ĠĠ(9300)^4Ġ)Ġ/Ġ(7
Rollout 98, Act: -1.731
ĠlongerĠlifetime.ĠTheĠWĠVirginisĠstarĠisĠaĠPopulationĠIIĠCepheid,ĠwhichĠindicatesĠit'sĠolder
Rollout 98, Act: -1.753
ĠwhileĠ**G2V**Ġ(Sun-like,ĠPopulationĠI)ĠandĠ**M4V**Ġ(red
Rollout 21, Act: -1.793
ĠwithĠvariablesĠx1,Ġx2,Ġetc.,ĠapplicationsĠofĠf,Ġaddition,Ġmultiplication,Ġsubtraction,Ġand
Rollout 98, Act: -1.846
ĠisĠinĠaĠsystemĠwithĠaĠG2VĠ(PopulationĠI),ĠwouldĠthatĠbeĠpossible?ĠMaybeĠinĠthe

DOWN_PROJ

Top 16 Positive Activations
Rollout 89, Act: 1.522
ĠproblemĠisĠaltered,Ġthus:ĊĊSupposeĠthere'sĠmisprintĠinĠtheĠproblem.ĠInsteadĠ2/5
Rollout 99, Act: 1.515
ĠevenĠwithĠtheĠtwoĠiterations,ĠbutĠdueĠtoĠpossibleĠmisconfusionĠinĠtheĠproblemĠstatement.ĊĊAlternatively,Ġmaybe
Rollout 84, Act: 1.482
/9ĠasĠmuchĠbeverage".ĊĊAlternatively,ĠsomeoneĠcouldĠmisreadĠitĠasĠpurchasingĠhalfĠasĠmuch,ĠandĠthen
Rollout 84, Act: 1.469
ĠasĠmuchĠandĠdrunkĠtwiceĠasĠmuch".ĠSomeĠmightĠmisinterpretĠ"twiceĠasĠmuch"ĠasĠtwiceĠthe
Rollout 95, Act: 1.419
ĠnotĠequal.ĠTherefore,ĠtheĠproblemĠisĠflawedĠorĠmisinterpretationĠisĠthere.ĊĊĠWait,ĠbutĠproblemĠstates
Rollout 84, Act: 1.414
ĠtimeĠthroughĠtheĠproblemĠtoĠmakeĠsureĠIĠdidn'tĠmisinterpretĠanything.ĊĊTheĠproblemĠsaysĠ"heĠhadĠpurchased
Rollout 95, Act: 1.401
AreaĠABC,ĠwhichĠareĠnotĠconstants.ĊĊĠUnlessĠsomeĠmiscomputationĠiĠmade.ĊĊĠLet'sĠrecheckĠarea
Rollout 89, Act: 1.374
/5thĠregardlessĠofĠboys.ĊĊWait,ĠthatĠmisinterpretation.ĊĊWait,ĠIfĠsomeoneĠreadsĠitĠas
Rollout 64, Act: 1.372
ĠbeĠlikeĠthat.ĊĊPerhapsĠthere'sĠaĠmistakeĠinĠmisapplyingĠtheĠinitialĠformula.ĊĊHoldĠon.ĠLet
Rollout 89, Act: 1.360
ĠaĠwholeĠnumber.ĊĊUnlessĠtheĠtestĠquestionĠhasĠaĠmisprint,ĠandĠactuallyĠthereĠareĠ0.2
Rollout 30, Act: 1.355
ĠstepĠofĠtheĠreasoningĠbecauseĠmistakesĠinĠindexĠshiftsĠorĠmisapplyingĠtheĠrecursionsĠcouldĠaffectĠtheĠanswer.ĊĊ
Rollout 89, Act: 1.351
ĠtoĠinterpretĠthis?ĊĊWait,ĠmaybeĠtheĠproblemĠwasĠmisstated.ĊĊAlternatively,ĠmaybeĠtheĠoriginalĠstatementĠis
Rollout 41, Act: 1.340
,Ġright?ĊĊWait,ĠholdĠon.ĠMaybeĠIĠmisread.ĠTheĠquestionĠisĠ"atĠmostĠtwoĠdifferent
Rollout 83, Act: 1.329
ĠwouldĠfixĠtheĠarrangements,ĠsinceĠotherĠrotationsĠwouldĠcauseĠmisalignment.ĊĊButĠwait,ĠbutĠmaybeĠthisĠisĠincorrect
Rollout 89, Act: 1.288
thĠofĠchildrenĠareĠgirls."ĠThatĠcouldĠbeĠaĠmisinterpretation.ĠIfĠ"2/5thĠof
Rollout 99, Act: 1.276
Ġx4.ĠHowever,Ġlikely,ĠthereĠwasĠaĠmiscountĠinĠtheĠiterationĠsteps.ĊĊAlternatively,ĠperhapsĠMull
Top 16 Negative Activations
Rollout 84, Act: -1.048
ĠdrankĠm/nĠofĠ1Ġunit,ĠsoĠtheĠactualĠamountĠheĠdrankĠoriginallyĠwasĠm/nĠ*Ġ1
Rollout 37, Act: -1.052
2015!.ĠThenĠperhapsĠthroughĠcombinatorialĠinterpretations.ĊĊWait,ĠholdĠon.ĠLetĠmeĠconsider
Rollout 48, Act: -1.055
ĠbytesĠworthĠofĠheader.ĠThen,Ġpresumably,ĠtheĠactualĠIPĠpacketĠwithĠdataĠhasĠ52ĠbytesĠtotal
Rollout 37, Act: -1.057
ĠconsecutivelyĠassigned.ĠĠOrĠthroughĠcombinatorialĠinterpretations.ĠButĠIĠhaven'tĠseenĠthisĠbefore.
Rollout 24, Act: -1.062
ĠtoĠaĠkindĠofĠfixedĠpointĠinĠsomeĠcombinatorialĠstructure.ĊĊAnotherĠidea:ĠSupposeĠ(R1,
Rollout 93, Act: -1.067
ĠrememberĠthatĠsuchĠsumsĠcanĠbeĠevaluatedĠusingĠcombinatorialĠidentitiesĠorĠorthogonalityĠrelationsĠwithĠCheb
Rollout 24, Act: -1.073
variantsĠlikeĠrank.ĊĊAlternatively,ĠaĠpivotalĠcombinatorialĠlemma.ĊĊAlright,ĠperhapsĠit'sĠfinallyĠtimeĠto
Rollout 33, Act: -1.074
ĠmaybeĠthere'sĠaĠgeneratingĠfunctionĠorĠsomeĠcombinatorialĠtrickĠtoĠhandleĠit.ĊĊAlternatively,ĠperhapsĠweĠcan
Rollout 16, Act: -1.079
ĠweĠcanĠuseĠinclusion-exclusionĠorĠsomeĠcombinatorialĠapproach.ĊĊExpandingĠtheĠtermĠ[1Ġ-Ġ(
Rollout 26, Act: -1.091
ĠabsoluteĠterms,ĠÎĶ(n)*R*T.ĠTheĠactualĠvalueĠcomputedĠwasĠ-7.4202
Rollout 48, Act: -1.092
ĠwouldĠbeĠ0xFB12.ĠButĠtheĠactualĠvalueĠwouldĠthenĠbeĠFB12.ĠButĠif
Rollout 41, Act: -1.099
ĠtwoĠdifferentĠdigits?ĊĊThisĠisĠaĠclassicĠcombinatorialĠquestion.ĊĊNumberĠofĠsuchĠnumbersĠ=ĠC(1
Rollout 24, Act: -1.105
ĠthereĠprobablyĠisĠaĠtheoremĠforĠthisĠinĠcombinatorialĠgameĠtheoryĠorĠhypergraphsĠthatĠIĠdon'tĠremember
Rollout 26, Act: -1.110
=Ġ-180.76Ġ(precise),ĠbutĠgivenĠpossibleĠuncertainties:ĊĊIfĠweĠconsiderĠall
Rollout 81, Act: -1.112
ĠdifferentĠequations.ĠHowever,ĠthisĠisĠnotĠpossibleâĢĶtheĠactualĠmathematicalĠrelationshipĠmustĠconsistent.ĠSoĠIĠneedĠtoĠfigure
Rollout 10, Act: -1.154
ĠcalledĠlosingĠpositionsĠorĠP-positionsĠinĠcombinatorialĠgameĠtheory.ĠTheĠproblemĠstatesĠthereĠareĠinfinitelyĠmany

Layer 3

GATE_PROJ

Top 16 Positive Activations
Rollout 88, Act: 0.544
ĠisĠ\(\boxed{100}\).<|im_end|>
Rollout 91, Act: 0.505
|\)ĠisĠ\(\boxed{2}\).<|im_end|>
Rollout 31, Act: 0.499
-\sqrt{5}}{2}}\).<|im_end|>
Rollout 99, Act: 0.492
boxed{2.157}Ġ\).<|im_end|>
Rollout 32, Act: 0.479
Ġ\(\boxed{4018}\).<|im_end|>
Rollout 0, Act: 0.464
ĠisĠ\(\boxed{128}\).<|im_end|>
Rollout 83, Act: 0.426
ĠisĠ\(\boxed{144}\).<|im_end|>
Rollout 10, Act: 0.391
ĠwhereĠtheĠsecondĠplayerĠcanĠforceĠaĠwin.}<|im_end|>
Rollout 27, Act: 0.343
^{6}Ġ\text{ĠK}}\).<|im_end|>
Rollout 95, Act: 0.328
ĠrationalĠlengthĠisĠ\(\boxed{0}\).<|im_end|>
Rollout 53, Act: 0.312
dfrac{27}{2}}\).<|im_end|>
Rollout 1, Act: 0.297
ĠyĠ\text{ĠexistsĠasĠdescribed}}Ċ\]<|im_end|>
Rollout 65, Act: 0.289
)ĠisĠ\(\boxed{40}\).<|im_end|>
Rollout 72, Act: 0.284
{\text{AllĠevenĠpositiveĠintegers}}Ċ\]<|im_end|>
Rollout 67, Act: 0.275
ĠisĠ\(\boxed{348}\).<|im_end|>
Rollout 94, Act: 0.265
ĠcorrectĠanswerĠisĠ\(\boxed{B}\).<|im_end|>
Top 16 Negative Activations
Rollout 48, Act: -5.014
,ĠthenĠtakeĠtheĠone'sĠcomplement.ĊĊButĠlet'sĠtryĠthisĠstepĠbyĠstep:ĊĊAddĠallĠtheĠ
Rollout 90, Act: -5.019
ĠrelateĠN,ĠL,ĠandĠM.ĊĊButĠlet'sĠstartĠwithĠ9NĠ+ĠMĠâī¡Ġ
Rollout 40, Act: -5.021
Ġ-ĠÏī^{150})ĊĊButĠlet'sĠfirstĠcomputeĠÏī^{150}.ĠSince
Rollout 56, Act: -5.024
1,ĠA2,ĠA3?ĊĊButĠlet'sĠmakeĠsure:ĠtheĠoriginalĠtriangleĠwasĠA0A
Rollout 91, Act: -5.031
Ġ=Ġ-1.ĠSo,Ġposition.ĊĊButĠlet'sĠdefineĠcoordinates.ĠLet'sĠtakeĠDĠasĠtheĠorigin
Rollout 39, Act: -5.033
ĠtermsĠhere.ĠHmm,Ġokay.ĊĊSo,Ġlet'sĠsee.ĠIfĠIĠneedĠtoĠcomputeĠtheĠsumĠof
Rollout 39, Act: -5.034
Ġ7101.ĊĊWait,ĠbutĠlet'sĠthinkĠaboutĠit.Ġ10^nĠ-Ġk
Rollout 17, Act: -5.043
0.0254ĠTesla.ĊĊButĠlet'sĠalsoĠconfirmĠcalculations:ĊĊGiven:ĊĊμ0=4
Rollout 35, Act: -5.043
ĠisĠaĠdivisorĠofĠ10.ĊĊButĠlet'sĠlistĠallĠpossibleĠdivisorsĠofĠ10:
Rollout 1, Act: -5.043
ĠcouldĠbeĠuncountable.ĠHmm.ĊĊButĠlet'sĠthink:ĠSupposeĠthatĠSĠisĠaĠsetĠwhereĠall
Rollout 78, Act: -5.046
ĠtrainĠtoĠgoĠpastĠSteve.ĊĊWait,ĠbutĠlet'sĠtakeĠthisĠstepĠbyĠstep.ĊĊFirst,Ġlet's
Rollout 99, Act: -5.052
âīĪ2.1573.ĊĊButĠlet'sĠcheckĠifĠperhapsĠmyĠinitialĠassumptionĠofĠcoefficientsĠisĠwrong
Rollout 98, Act: -5.059
.ĠTheĠanswerĠisĠBĠ(3).ĊĊButĠlet'sĠjustĠcheckĠforĠsystemĠ1ĠonceĠmore.ĠIf
Rollout 35, Act: -5.059
ĠtoĠdetermineĠn1ĠandĠn2.ĊĊButĠlet'sĠdiveĠaĠbitĠdeeper.ĠSinceĠQ(n)Ġ=
Rollout 23, Act: -5.088
ĠsoĠyes,ĠmostĠhaveĠdecayed.ĊĊButĠlet'sĠdouble-check.ĠForĠe^-73.0
Rollout 52, Act: -5.114
Ġnegative.ĠButĠweĠneedĠtoĠsee.ĊĊButĠlet'sĠtakeĠtheĠnextĠstep.ĊĊFirst,ĠfactorizingĠ

UP_PROJ

Top 16 Positive Activations
Rollout 28, Act: 1.236
(a,0),ĠB(0,b),ĠG(a/3,Ġb/3).ĠLet'sĠtake
Rollout 9, Act: 1.184
ĠcommonĠtoĠtriangleĠ$GEM$ĠandĠsquareĠ$AIME$ĠisĠ$80$ĠsquareĠunits.
Rollout 12, Act: 1.177
)Ġ\cdotĠb'(\nu)Ġ-ĠF(a(\nu),Ġ\nu)Ġ\cdotĠa'
Rollout 28, Act: 1.154
Now,ĠpointsĠO(0,0),ĠA(a,0),ĠB(0,b),ĠG(a
Rollout 28, Act: 1.132
(a,0),ĠB(0,b),ĠG(a/3,Ġb/3Ġ).ĊĊThisĠisĠthe
Rollout 77, Act: 1.131
Ġ=Ġ[(aĠ+Ġb)/b]Ġ*Ġf(a,ĠbĠ-Ġa)Ġ...ĠWait,ĠholdĠon
Rollout 59, Act: 1.113
ĠcoinsĠmustĠbeĠtails.ĠSoĠtheĠprobabilityĠisĠP(A=T)P(B=T)P(C=T)
Rollout 96, Act: 1.112
5}{12}$ĠthatĠcontainsĠtheĠpointĠ$AĠ=Ġ(24,-1)$Ġ,ĠandĠlet
Rollout 56, Act: 1.106
_{0}$ĠandĠtheĠnewĠpastureĠisĠtriangleĠ$A_{1}ĠA_{2}ĠA_{3
Rollout 28, Act: 1.105
Ġsystem,ĠwithĠO(0,0),ĠA(a,0),ĠB(0,b),ĠcenteredĠat
Rollout 30, Act: 1.105
ĠsatisfyĠ$a_{n+1}=\left(a_{n-1}-1\right)\left
Rollout 28, Act: 1.104
InĠoriginalĠcoordinatesĠO(0,0),ĠA(a,0),ĠB(0,b),ĠG(a
Rollout 77, Act: 1.102
Ġ-Ġa)Ġ=Ġ(bĠ-Ġa)Ġf(a,ĠaĠ+Ġ(bĠ-Ġa))Ġ=>Ġb
Rollout 56, Act: 1.090
n}ĠA_{n-2}$ĠtoĠ$A_{n+1}$ĠsuchĠthatĠ$A_{
Rollout 77, Act: 1.087
Ġ=Ġ[bĠ/Ġ(bĠ-Ġa)]Ġf(a,ĠbĠ-Ġa)ĊĊButĠthisĠstillĠinvolvesĠf
Rollout 77, Act: 1.082
Ġ-Ġa)Ġ=Ġ(bĠ-Ġa)Ġf(a,Ġb)ĊĊTherefore:ĊĊf(a,Ġb)
Top 16 Negative Activations
Rollout 79, Act: -2.075
ĠButĠnot.ĠAsĠbefore.ĊĊWait!ĠAĠcommonĠtermĠthatĠmeansĠinvestigators'ĠfacilityĠisĠ"OperationsĠCenter".
Rollout 25, Act: -2.075
.ĠThus,ĠwithĠveryĠlargeĠμ_final,ĠperigeeĠcanĠbeĠmadeĠasĠsmallĠasĠdesired.ĠHowever
Rollout 25, Act: -2.076
final}}Ġ\)**:ĊĠĠĠUsingĠtheĠdesiredĠperigeeĠdistanceĠ\(Ġr_p_{\text{final
Rollout 25, Act: -2.076
Ġ\(ĠrĠ\)**:ĊĠĠĠUsingĠtheĠoriginalĠperigeeĠdistanceĠ\(Ġr_p_{\text{initial
Rollout 56, Act: -2.099
ĠstepĠaddsĠhalfĠofĠtheĠpriorĠamount.ĊĊSoĠgeneralĠterm:ĠareaĠS_n=K_n=Ġ(sum
Rollout 65, Act: -2.103
,ĠwhichĠisĠsameĠasĠoriginal.ĊĊAlternatively,Ġnon-correspondingĠratios.ĠIfĠtheĠnewĠprismĠhasĠsides
Rollout 46, Act: -2.124
ĠholdĠon.ĠLetĠmeĠclarify:ĊĊTheĠcrossĠproductĠtermĠÏīĠÃĹĠv,ĠinĠcomponentsĠassumingĠÏī
Rollout 10, Act: -2.224
5,ĠtheĠGrundyĠnumbersĠmayĠbeĠnon-periodic.ĊĊButĠsinceĠmanualĠcalculationĠisĠtedious,Ġlet's
Rollout 25, Act: -2.239
,ĠraisingĠmuĠfurtherĠseemsĠpossible.ĊĊButĠasĠperigeeĠformula:ĊĊr_pĠ=Ġ[h2/m
Rollout 10, Act: -2.252
ĠweĠcanĠfindĠaĠdifferentĠpatternĠorĠaĠnon-periodicĠstructure.ĊĊStartingĠfromĠn=0Ġ(L),
Rollout 83, Act: -2.294
Ġfixed;ĠsinceĠtheĠdigitsĠareĠunique,ĠtheĠautomorphismsĠareĠonlyĠrotations,ĠandĠanyĠrotatedĠcopyĠdiffers
Rollout 15, Act: -2.306
ĠmatrixĠwithĠ1sĠonĠtheĠsubĠandĠsuperdiagonals.ĊĊForĠinstance,ĠifĠyouĠhaveĠthis
Rollout 10, Act: -2.311
)ĠwereĠcomputedĠforĠseveralĠpositions,ĠshowingĠnon-periodicĠbutĠstructuredĠbehavior.ĊĊ4.Ġ**ConstructingĠInfinite
Rollout 15, Act: -2.325
ĠwouldĠhaveĠaĠsingleĠ1ĠonĠtheĠfirstĠsuperdiagonalĠandĠfirstĠsubdiagonal.ĊĊWait,Ġexample
Rollout 87, Act: -2.540
5ĊĊOtherĠcombinations.ĠForĠinstance,ĠconsideringĠhypergeometricĠfunctionsĠorĠsimilar,ĠthereĠisĠnoĠclosedĠform
Rollout 12, Act: -2.588
.ĠNotĠexactlyĠmatchingĠhere.ĊĊAlternatively,ĠconsiderĠhypergeometricĠsubstitutions.ĊĊWait,ĠbutĠmaybeĠthat'sĠgetting

DOWN_PROJ

Top 16 Positive Activations
Rollout 35, Act: 1.497
Ġ=Ġ\boxed{418}Ċ\]<|im_end|>
Rollout 31, Act: 1.480
-\sqrt{5}}{2}}\).<|im_end|>
Rollout 19, Act: 1.474
\[Ċ\boxed{78}Ċ\]<|im_end|>
Rollout 72, Act: 1.465
{\text{AllĠevenĠpositiveĠintegers}}Ċ\]<|im_end|>
Rollout 59, Act: 1.448
[Ċ\boxed{515}Ċ\]<|im_end|>
Rollout 5, Act: 1.396
[Ċ\boxed{181}Ċ\]<|im_end|>
Rollout 81, Act: 1.368
(angleĠBAC)=22/7ĠTherefore,ĠdespiteĠformulaĠinĠsecondĠapproachĠappearsConflicting,ĠtheĠcoordinate
Rollout 50, Act: 1.365
[Ċ\boxed{112}Ċ\]<|im_end|>
Rollout 69, Act: 1.362
[Ċ\boxed{204}Ċ\]<|im_end|>
Rollout 60, Act: 1.350
]ĊĊ\[Ċ\boxed{6}Ċ\]<|im_end|>
Rollout 99, Act: 1.350
boxed{2.157}Ġ\).<|im_end|>
Rollout 91, Act: 1.349
|\)ĠisĠ\(\boxed{2}\).<|im_end|>
Rollout 80, Act: 1.339
\[Ċ\boxed{11}Ċ\]<|im_end|>
Rollout 76, Act: 1.338
[Ċ\boxed{929}Ċ\]<|im_end|>
Rollout 68, Act: 1.335
[Ċ\boxed{501}Ċ\]<|im_end|>
Rollout 16, Act: 1.329
{789953}Ċ\]<|im_end|>
Top 16 Negative Activations
Rollout 5, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊOneĠbaseĠofĠaĠtrapezoidĠis
Rollout 34, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊFindĠcontinuousĠfunctionsĠĠ$x(t),\
Rollout 47, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠtrainĠwithĠcross-sectionalĠareaĠ$S_{
Rollout 43, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠC-NOTĠgateĠwhereĠthe
Rollout 60, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequencyĠ$f
Rollout 55, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĊ\[\mathbf{A}
Rollout 63, Act: -1.250
systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$P$ĠĠdenoteĠtheĠset
Rollout 26, Act: -1.286
s,ĠsoĠgivenĠconflictingĠrules.ĊĊHowever,ĠinĠmostĠsuchĠproblems,ĠevenĠwithĠintermediateĠprecision,ĠtheĠanswer
Rollout 23, Act: -1.289
,åı¯èĥ½æľĥéģ¸æĵĩD.ĊĊButĠæł¹æĵļé¡Į缮,çŃĶæ¡Īä¸įåºĶ该æĺ¯total,èĢĮæĺ¯alphaçļĦãĢĤ
Rollout 57, Act: -1.296
ĠofĠ\(Ġ\complementĠUĠ\)Ġ(denotedĠ\(ĠVĠ\)).ĠEveryĠpointĠinĠ\(ĠCĠ\
Rollout 11, Act: -1.312
ĠofĠtheĠformĠtĠ+1/t.ĊĊThus,ĠdenoteĠt1Ġ=a,Ġt2=ĠaĠz
Rollout 57, Act: -1.375
**:ĠTheĠcomplementĠofĠ\(ĠUĠ\)Ġ(denotedĠ\(Ġ\complementĠUĠ\))ĠisĠclosed,
Rollout 25, Act: -1.467
ĠbeĠsmaller,ĠmeaningĠitĠcouldĠapproachĠcloserĠbeforeĠbreakingĠapart.ĠHowever,ĠtheĠquestionĠisĠaboutĠmakingĠcontactĠwith
Rollout 11, Act: -1.873
ĠnotĠonĠunitĠcircleĠeither.ĊĊAdditionally,Ġ(denoteĠSĠ=ĠaĠ+ĠbĠ+Ġc,ĠTĠ=
Rollout 24, Act: -2.248
ĠR2ĠdominatingĠr1ĠoverĠC2.ĠDenoteĠthisĠrowĠasĠf(r1)ĠâĪĪĠR2
Rollout 1, Act: -2.381
ĠintoĠequationsĠinvolvingĠt.ĠLetĠmeĠsee.ĊĊDenoteĠtĠ=Ġ2y.ĠThenĠyĠ=Ġt

Layer 4

GATE_PROJ

Top 16 Positive Activations
Rollout 0, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 1, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 2, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 3, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 4, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 11, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 6, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 7, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 8, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 9, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 10, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 16, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 12, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 13, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 14, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 15, Act: 0.012
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Top 16 Negative Activations
Rollout 21, Act: -6.294
âĨĴZĠwithĠsuchĠregularityĠ(likeĠsatisfyingĠadditivityĠandĠcertainĠfunctionalĠequations)ĠmightĠhaveĠtoĠbeĠconstant
Rollout 49, Act: -6.295
ĠhasĠcountableĠspreadĠandĠisĠT5Ġ(hereditaryĠnormal),ĠwhichĠweĠmightĠorĠmightĠnotĠhave,
Rollout 46, Act: -6.298
ĠwithĠangularĠvelocityĠÏī.ĠTheĠaccelerationĠinĠtheĠinertialĠframeĠisĠa_inertialĠ=Ġa_rot
Rollout 21, Act: -6.300
ĠtheĠimageĠisĠtheĠentireĠsetĠZ.ĠSoĠsurjectiveĠontoĠZ.ĠSoĠfunctionsĠinĠXĠareĠsurjective
Rollout 46, Act: -6.307
,ĠwhichĠareĠusuallyĠpartĠofĠtheĠpotentialĠinĠanĠinertialĠframe.ĠTheĠlastĠtermĠisĠsimilarĠtoĠa
Rollout 81, Act: -6.333
Ġconfusing;ĠangleĠBACĠisĠjustĠoneĠangleĠatĠvertexĠA,ĠbutĠperhapsĠinĠtermsĠofĠcoordinateĠgeometry,
Rollout 21, Act: -6.335
1},ĠwhichĠareĠsubsetsĠofĠZ,ĠnotĠsurjectiveĠontoĠZ.ĠSoĠSĠwouldĠnotĠbeĠaĠsubset
Rollout 21, Act: -6.336
ĠholdsĠforĠtheĠidentityĠfunctionĠisĠtrivial.ĠForĠsurjectiveĠfunctions,ĠinjectiveĠfunctions,ĠsinceĠ2f
Rollout 72, Act: -6.338
:ĊĊInĠaĠregularĠpentagon,ĠtheĠdiagonalĠoverĠsideĠisĠtheĠgoldenĠratioĠÏĨĠ=Ġ(1Ġ+Ġsqrt
Rollout 1, Act: -6.345
CompareĠtoĠtheĠaboveĠline:ĊĊFromĠtheĠsumĠofĠnorms,ĠweĠhave:ĊĊ-⣨xĠ+Ġx',
Rollout 21, Act: -6.372
.ĠButĠtoĠhaveĠimageĠexactlyĠZ,ĠtheĠsurjective.ĠIfĠaĠfunctionĠfĠsatisfiesĠf(xĠ-f
Rollout 46, Act: -6.417
ĠtoĠtheĠgradientĠofĠU?ĠButĠonlyĠinĠanĠinertialĠframeĠwouldĠtheĠaccelerationĠequalĠtheĠgradientĠofĠthe
Rollout 46, Act: -6.431
?ĠHowever,ĠthatĠwouldĠbeĠtheĠcaseĠinĠanĠinertialĠframe,ĠbutĠtheĠpresenceĠofĠtheĠcentrifugal
Rollout 46, Act: -6.432
ĊĊButĠtheseĠwouldĠcorrespondĠtoĠtheĠaccelerationĠinĠanĠinertialĠframeĠminusĠtheĠcentrifugalĠacceleration?ĠI'm
Rollout 46, Act: -6.495
,ĠwithoutĠconsideringĠtheĠrotatingĠframe,ĠsoĠassumingĠanĠinertialĠframeĠwhereĠUĠisĠtheĠconventionalĠgravitationalĠpotential,
Rollout 84, Act: -6.614
ĠandĠ"twiceĠasĠmuch"ĠareĠbothĠadverbialĠphrasesĠmodifyingĠtheĠverbsĠ"purchased"Ġand

UP_PROJ

Top 16 Positive Activations
Rollout 7, Act: 3.364
ĠĠ$47$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'mĠtryingĠtoĠsolveĠthis
Rollout 44, Act: 3.314
mathrm{T}_{2}$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠwhich
Rollout 26, Act: 3.305
mol}^{-1}$Ġ]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthisĠproblem
Rollout 98, Act: 3.256
Ċ(D)Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠhow
Rollout 6, Act: 3.238
$Ġ.ĠĠ[/*]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthese
Rollout 94, Act: 3.231
Ċ(D)Ġa,bĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠdetermineĠwhichĠof
Rollout 18, Act: 3.215
=H_{xy}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblemĠstepĠby
Rollout 45, Act: 3.193
ĠofĠĠ$Ġm$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠproblemĠwhereĠa
Rollout 36, Act: 3.189
Ġ.ĠFindĠ$AP.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠhow
Rollout 29, Act: 3.182
Ġ\rhoĠgH$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 23, Act: 3.180
555ĠBqĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠstepĠby
Rollout 79, Act: 3.179
ĠfacilityĠ(8,4)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠcrosswordĠclue.
Rollout 54, Act: 3.174
,ĠBĠ=Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestionĠstepĠby
Rollout 1, Act: 3.172
mathcal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠthere
Rollout 11, Act: 3.169
Ġ.ĊĊ*DavidĠAltizio*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠproblemĠhereĠwhere
Rollout 55, Act: 3.157
018}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠmatrixĠAĠand
Top 16 Negative Activations
Rollout 22, Act: -2.270
T/m)ĊĊv_meanĠ=Ġsqrt(8kT/(ÏĢm))ĊĊAlright,ĠsoĠfirstĠIĠneed
Rollout 54, Act: -2.275
ideĠ(strongĠbase)ĊĊ5.ĠEthanethiolateĠ(weakerĠbaseĠthanĠhydroxide)ĊĊ3
Rollout 15, Act: -2.280
Ġ-ĠTheĠsystemĠcanĠbeĠrepresentedĠbyĠaĠtridiagonalĠmatrixĠ\(ĠMĠ\)ĠwhereĠeachĠrowĠhasĠ
Rollout 54, Act: -2.291
ĠisĠmostĠhydrated)Ċ-ĠRSâĪĴĠ(thiolate)ĠisĠaĠbetterĠnucleophileĠthanĠROâĪĴĠ(
Rollout 54, Act: -2.291
,ĠmethanolĠ(neutral),Ġethanethiolate.ĠPropionateĠisĠaĠcarboxylate,
Rollout 54, Act: -2.294
KaĠaroundĠ10,ĠethanethiolateĠ(R-SâĪĴ)ĠwouldĠhaveĠaĠbaseĠstrength
Rollout 54, Act: -2.302
ĠbaseĠ(strongestĠacid),ĠethanethiolateĠnextĠ(stillĠweakerĠbaseĠthanĠhydroxideĠand
Rollout 26, Act: -2.307
DeltaĠH_f^\circĠ(\mathrm{HgO})Ġ=Ġ\frac{-180.7
Rollout 97, Act: -2.315
ĠmayĠnotĠbeĠprecise.ĠBecauseĠtheĠreactionĠstoichiometryĠisĠmoles,ĠbutĠinĠtermsĠofĠpartialĠpressures
Rollout 54, Act: -2.340
Ġbases.ĊĊThus,ĠevenĠthoughĠethanethiolateĠ(5)ĠisĠaĠweakerĠbaseĠthanĠhydrox
Rollout 54, Act: -2.345
)Ġ>ĠhydroxideĠ>ĠethanethiolateĠ>ĠpropionateĠ(weakestĠbase).ĊĊThen,
Rollout 54, Act: -2.345
:ĊĊOrder:Ġ5Ġ(ethanethiolate)Ġ>Ġ2Ġ(hydroxide)Ġ>
Rollout 54, Act: -2.410
Ġsolution:ĊĊInĠSN2Ġreactions,ĠtheĠthiolateĠionĠ(e.g.,ĠSHâĪĴ)ĠisĠa
Rollout 22, Act: -2.426
Ġare:ĊĊv_rmsĠ=Ġsqrt(3kT/m)ĊĊv_meanĠ=Ġsqrt(8kT
Rollout 15, Act: -2.537
SuchĠmatricesĠareĠcalledĠJacobiĠmatricesĠorĠtridiagonalĠmatrices.ĠNilpotencyĠcanĠbeĠdiscussedĠviaĠtheir
Rollout 15, Act: -2.596
ĠpolynomialĠisĠx^n.ĊĊButĠforĠanyĠtridiagonalĠmatrix,ĠparticularlyĠsymmetricĠones,ĠeigenvaluesĠareĠreal

DOWN_PROJ

Top 16 Positive Activations
Rollout 79, Act: 2.599
Ġuse:ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.588
ĠâĢĵĠlettersĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.533
Ġchanges":ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.529
Ġchanges":ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.476
Ġagain:ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.465
"Ġ:ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.461
Ġchanges":ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.446
ĠlettersĠ(M,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.435
Ġletters:ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.414
ĠlettersĠ(M,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.409
Ġletters:ĠM,ĠI,ĠN,ĠO,ĠR,ĠH,ĠN,ĠG.ĠLet'sĠsee
Rollout 79, Act: 2.395
Ġcheck:ĠM,ĠI,ĠN,ĠO,ĠRĠplusĠH,ĠG,ĠN.ĠThatĠcouldĠbe
Rollout 79, Act: 2.385
ĠhasĠlettersĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.375
Ġletters:ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.373
ĠhaveĠareĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Rollout 79, Act: 2.336
)ĠareĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG
Top 16 Negative Activations
Rollout 79, Act: -1.375
".ĊĊSoĠmaybeĠ"minorĠchanges"ĠisĠanĠanagramĠforĠtheĠanswer.ĠLet'sĠtryĠtoĠrearrange
Rollout 49, Act: -1.380
ĠunionĠofĠclosedĠsets)ĠisĠcalledĠaĠ"perfectlyĠnormal"Ġspace?ĠNotĠexactly.ĠPerfectlyĠnormal
Rollout 79, Act: -1.382
ĠanswerĠisĠ"somethingĠsomething"Ġthat'sĠanĠanagramĠofĠ"minorĠchanges"ĠplusĠsomeĠletters.ĠWait
Rollout 79, Act: -1.382
.ĠSoĠmaybeĠ"minorĠchanges"ĠisĠtheĠanagramĠfodder,ĠwhichĠwouldĠformĠanĠ8-letterĠword
Rollout 79, Act: -1.388
".ĠTheĠwordplayĠusuallyĠinvolvesĠsomeĠkindĠofĠanagram,Ġreversal,ĠhiddenĠword,Ġetc.Ġ"Not
Rollout 39, Act: -1.390
9...ĠpartĠbecomesĠ111...ĠandĠthenĠ7101ĠbecomesĠ789
Rollout 79, Act: -1.397
Ġagain.ĊĊIfĠ"Noticed"ĠisĠtheĠanagramĠindicator,ĠthenĠanagramĠofĠ"minorĠchangesĠin
Rollout 79, Act: -1.403
'sĠsee.ĠIfĠ"noticed"ĠisĠtheĠanagramĠindicator,ĠthenĠ"minorĠchanges"ĠhasĠtoĠbe
Rollout 51, Act: -1.424
ĠvariableĠtoĠyĠ=ĠxĠ-Ġθ,ĠandĠthenĠconsideringĠtheĠintegralĠfromĠ-θĠtoĠ2ÏĢ
Rollout 79, Act: -1.427
,ĠthenĠ"minorĠchanges"ĠhasĠtoĠbeĠanagrammed.ĠLet'sĠcheckĠtheĠletters.Ġ"Minor
Rollout 74, Act: -1.456
Ġfore-and-aftĠcomponentĠK_bĠ(proportionalĠtoĠtheĠEarth'sĠfield?)ĠandĠathwartships
Rollout 57, Act: -1.462
penĠsets.ĠWait,ĠperhapsĠregularityĠisĠaĠresultĠofĠtheĠdensityĠtopologyĠbeingĠaĠregularĠtopologyĠthroughĠits
Rollout 79, Act: -1.483
Ġetc.Ġ"Noticed"ĠmightĠbeĠtheĠanagramĠindicator.Ġ"MinorĠchanges"ĠcouldĠbeĠtheĠletters
Rollout 62, Act: -1.532
âĪĴm+11.](https://oeis.org/search?q=%22m%5E
Rollout 79, Act: -1.571
ĠwordsĠfromĠtheseĠlettersĠinclude:ĊĊ-Ġ"Hernias"ĠplusĠsomeĠothers.ĠNo.ĠMaybeĠ"Chair
Rollout 23, Act: -1.590
YouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊImagineĠaĠradioactiveĠnucleiĠX(Z,A)ĠcanĠdecayĠinto

Layer 5

GATE_PROJ

Top 16 Positive Activations
Rollout 34, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 47, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 63, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 55, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 84, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 78, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 5, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 60, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 43, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 94, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 10, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 15, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 11, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 12, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 13, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 14, Act: 0.470
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Top 16 Negative Activations
Rollout 6, Act: -6.376
,Ġno,ĠbecauseĠtheĠmachineĠMĠdoesn'tĠhaveĠpriorĠknowledgeĠofĠα_n.ĠTheĠmachineĠMĠneedsĠto
Rollout 28, Act: -6.376
ĠO,ĠA,ĠB,ĠGĠareĠgivenĠwithĠcentroidĠG.ĠTheĠcentroidĠbeingĠ(a/3,
Rollout 11, Act: -6.379
ĠwhereĠdĠisĠsomeĠvariable.ĠButĠlikelyĠovercomplicating.ĊĊAlternatively,ĠassumeĠaĠ=Ġd,Ġb
Rollout 66, Act: -6.386
13,85)ĠisĠpairwiseĠcoprime.ĊĊTherefore,ĠmaybeĠthisĠisĠtheĠcorrectĠtriplet.
Rollout 81, Act: -6.404
ĠdegreesãĢĤĊĊConflictĠarises.ĠTheĠangleĠcalculatedĠhereĠwithĠcoordinateĠsystemĠsetupĠis约107Ġdegrees,butĠtan
Rollout 81, Act: -6.418
ĠAlternatively,ĠnoticeĠthatĠpreviousĠstepsĠgeneratedĠthisĠequationĠthroughĠcoordinateĠsystem.ĠInĠcoordinateĠsystemĠapproach,Ġh=5
Rollout 21, Act: -6.426
)Ġ-x))=0.ĠButĠperhapsĠoverĠcomplicating.ĊĊAlternatively,ĠsetĠequationsĠthatĠtakeĠfĠbeing
Rollout 1, Act: -6.433
ĠvectorsĠcouldĠformĠanĠorthogonalĠsetĠwithĠeachĠvectorĠhavingĠnormĠdĠ/Ġsqrt(2},ĠsoĠthatĠtheĠdistance
Rollout 95, Act: -6.434
ĠtĠfromĠ0Ġto1,ĠPkĠhasĠcoordinatesĠ(9t,Ġ0).ĠSimilarly,Ġpoint
Rollout 6, Act: -6.448
ĠnĠ+Ġn^kĠâīĪĠm?ĠUnlikely.ĠForĠexample,ĠmĠisĠdominatedĠbyĠn^
Rollout 96, Act: -6.450
,ĠandĠsimilarly,ĠanyĠpointĠonĠlineĠMĠhasĠcoordinateĠ(0,Ġδ).ĊĊButĠbecauseĠlinesĠLĠand
Rollout 24, Act: -6.462
ĠthisĠreasoningĠmayĠnotĠholdĠsinceĠtheĠdominationĠisĠoverĠcolumnsĠinĠC.ĊĊWait,ĠmaybeĠanotherĠangle.ĠFor
Rollout 96, Act: -6.510
ĠisĠtheĠorigin,ĠanyĠpointĠonĠlineĠLĠhasĠcoordinateĠ(γ,Ġ0),ĠwhereĠγĠisĠthe
Rollout 79, Act: -6.525
ĠthoseĠlettersĠmakeĠanĠ8-letterĠword?ĠUnlikely.ĠLet'sĠsee.ĠMaybeĠ"Scanning"?
Rollout 81, Act: -6.537
ĠapproachĠweĠcalculatedĠtheĠangleĠbetweenĠABĠandĠACĠwithĠcoordinateĠsystemĠwhereĠpointĠAĠisĠatĠ(0,Ġh
Rollout 38, Act: -6.613
ĠxĠisĠtheĠdistanceĠfromĠsoutheastĠcorner,ĠwhichĠhasĠcoordinateĠ(0,ĠW).ĠSoĠ(-x,ĠW

UP_PROJ

Top 16 Positive Activations
Rollout 69, Act: 1.960
12âĨĴa=Ġ-4.ĠButĠaĠmustĠbeĠaĠpositiveĠinteger,ĠsoĠcontradiction.ĊĊWhichĠsuggests
Rollout 12, Act: 1.939
ĠperiodĠdependingĠonĠaĠandĠb.ĠWait,ĠmoreĠprecisely,ĠforĠtheĠintegralĠinĠtheĠform:ĊĊâĪ«_
Rollout 65, Act: 1.900
ĠmultipleĠofĠ1995.ĠButĠaĠmustĠlessĠthanĠorĠequalĠtoĠ1995
Rollout 24, Act: 1.857
ĠtermsĠofĠdominanceĠonĠC,ĠandĠcolumnsĠinĠCĠmustĠ"cover"ĠallĠotherĠcolumnsĠinĠtermsĠofĠdominance
Rollout 67, Act: 1.832
=Ġ-15d/2.ĊĊButĠaĠmustĠdivideĠintoĠ1440ĠasĠperĠequation
Rollout 57, Act: 1.791
ĠyĠâĪīĠU).ĠĊĊWait,ĠbutĠmoreĠprecisely,ĠclosedĠsetsĠinĠtheĠdensityĠtopologyĠareĠpreciselyĠthe
Rollout 83, Act: 1.775
,4:Ġc,Ġa,Ġb.ĠSumĠmustĠbeĠ0.ĊĊContinuing,ĠsinceĠtheĠpattern
Rollout 97, Act: 1.771
Ġ(inĠtermsĠofĠpressure).ĠWait,ĠmaybeĠmoreĠprecisely:ĠBecauseĠtheĠdecompositionĠreducesĠN2O5Ġand
Rollout 29, Act: 1.758
'tĠbothĠbeĠinĠtheĠoptions,ĠperhapsĠtheĠanswerĠisĠE.ĊĊWait,ĠbutĠgivenĠthatĠPĠ=Ġrho
Rollout 65, Act: 1.756
*5*7*19.ĠSoĠaĠmustĠbeĠdivisibleĠbyĠ3,5,7,
Rollout 1, Act: 1.755
ĠwhereĠeachĠpairĠisĠseparatedĠbyĠd,ĠperhapsĠSĠmustĠlieĠonĠsuchĠaĠquadraticĠmanifold.ĠButĠunlessĠI
Rollout 97, Act: 1.740
ĠhaveĠtoĠbeĠcareful.ĊĊWait,ĠperhapsĠtheĠanswerĠisĠtoĠmodelĠtheĠpartialĠpressureĠofĠN2O5
Rollout 65, Act: 1.736
19=1995.ĠSoĠaĠmustĠbeĠaĠmultipleĠofĠ1995.
Rollout 81, Act: 1.731
ĠIĠneedĠtoĠrecallĠtheĠformula.ĠWait,ĠmoreĠprecisely,ĠinĠaĠgeneralĠtriangle,ĠtanĠAĠcanĠbe
Rollout 46, Act: 1.727
'sĠnotĠstrictlyĠ"inĠtermsĠofĠU".ĠIĠmustĠmisunderstandĠsomething.ĠAlternatively,ĠmaybeĠit'sĠaskingĠto
Rollout 65, Act: 1.725
/k.ĠSinceĠcĠmustĠbeĠinteger,ĠandĠaĠmustĠbeĠinteger:ĠaĠ=k*199
Top 16 Negative Activations
Rollout 87, Act: -2.319
ĠandĠanyĠnecessaryĠnumericalĠconstants.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠtheĠelectric
Rollout 74, Act: -2.323
,ĠandĠ$\theta$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠderiveĠanĠexpression
Rollout 16, Act: -2.323
ĠbyĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠEmiliaĠisĠtryingĠtoĠgrow
Rollout 10, Act: -2.327
ĠmatterĠhowĠhisĠopponentĠplays.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠproveĠthatĠin
Rollout 8, Act: -2.337
Ġ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠWeĠneedĠto
Rollout 58, Act: -2.341
ĠbyĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠSoĠPatchou
Rollout 93, Act: -2.360
2014}$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠproblemĠhere:
Rollout 79, Act: -2.371
ĠfacilityĠ(8,4)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠcrosswordĠclue.
Rollout 62, Act: -2.371
ĠfourĠnotĠnecessarilyĠdistinctĠprimes.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblem:ĠFind
Rollout 98, Act: -2.374
Ċ(D)Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠhow
Rollout 69, Act: -2.384
ĠTeamĠRoundĠ#10*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠpositiveĠintegers
Rollout 72, Act: -2.393
Ġ$ĠĊ1.Ġ$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblem:ĠFind
Rollout 42, Act: -2.403
oct-5-eneĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠthe
Rollout 83, Act: -2.403
ĠmeĠtryĠtoĠtackleĠthisĠproblem.ĠIt'sĠaboutĠlabelingĠtheĠverticesĠofĠaĠregularĠnonagonĠ(9-sided
Rollout 42, Act: -2.694
AnswerĠChoices:Ċ(A)Ġ[1,1'-bi(cyclobutan)]-1-eneĊ
Rollout 42, Act: -2.776
ĠoptionsĠare:ĊĊ(A)Ġ[1,1'-bi(cyclobutan)]-1-eneĠâĢĵ

DOWN_PROJ

Top 16 Positive Activations
Rollout 58, Act: 2.957
ĊOkay,Ġlet'sĠsee.ĠSoĠPatchouliĠhasĠanĠexamĠwithĠkĠparts,Ġright?ĠEach
Rollout 58, Act: 2.826
ĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊPatchouliĠisĠtakingĠanĠexamĠwithĠĠ$kĠ>Ġ
Rollout 37, Act: 2.598
ĠcouldĠsomehowĠseeĠthatĠeachĠtermĠinĠtheĠLeibnizĠformulaĠforĠdeterminantĠinvolvesĠsomeĠcommutators,
Rollout 34, Act: 2.473
ĠinsideĠtheĠintegrand,ĠweĠcanĠuseĠLeibniz'sĠrule.ĠTheĠgeneralĠformulaĠis:ĊĊd
Rollout 44, Act: 2.461
ĠisĠsimultaneouslyĠbothĠisothermalĠasĠwellĠasĠadiabaticĊĊ[C]ĠTheĠworkĠdoneĠbyĠtheĠgas
Rollout 12, Act: 2.458
ĠuseĠdifferentiationĠunderĠtheĠintegralĠsign,ĠprobablyĠLeibniz'sĠrule.ĠĊĊLeibniz's
Rollout 46, Act: 2.445
ĠaccelerationĠareĠbothĠincludedĠinĠU,ĠandĠthenĠallĠthat'sĠleftĠisĠtheĠCoriolisĠterm.ĠHmm
Rollout 15, Act: 2.416
ĠBecauseĠwhetherĠtheĠcountĠisĠoddĠorĠevenĠisĠallĠthatĠmatters.ĊĊWait:Ġexactly:ĊĊActually,ĠtoĠget
Rollout 42, Act: 2.415
ĠtheĠcarbocationĠundergoesĠaĠWagner-MeerweinĠrearrangement.ĠIfĠtheĠinitialĠcarbocationĠisĠat
Rollout 44, Act: 2.373
ĠbutĠifĠinsteadĠtheĠprocessĠisĠreversibleĠandĠadiabatic,ĠperhapsĠtheĠworkĠisĠdifferent.ĊĊWait,
Rollout 34, Act: 2.343
,ĠdsĠ\right)Ċ\]ĊĊUsingĠLeibniz'sĠrule,ĠweĠget:ĊĊ\[Ċx
Rollout 44, Act: 2.303
ĠV2).ĠOr,ĠifĠit'sĠadiabatic,Ġdifferent.ĊĊButĠtheĠquestionĠisĠcomparingĠthe
Rollout 3, Act: 2.296
ĠfocusĠmightĠspecificallyĠalignĠwithĠoneĠtemperature.ĊĊGivenĠallĠthat,ĠbutĠasĠperĠmyĠknowledgeĠcutoffĠisĠ2
Rollout 44, Act: 2.296
ĠtemperatureĠmightĠchange,ĠifĠit'sĠdoneĠadiabatically,Ġit'sĠaĠdifferentĠcalculation.ĊĊAlternatively,
Rollout 44, Act: 2.279
othermal,ĠthenĠless.ĠIfĠit'sĠadiabatic,Ġmore?ĊĊWait,ĠforĠexample,Ġlet
Rollout 17, Act: 2.240
0^-7ĠT·m/A.ĊĊGivenĠallĠthat,Ġlet'sĠuseĠcalculatorĠsteps:ĊĊComputeĠnumerator:
Top 16 Negative Activations
Rollout 32, Act: -4.342
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 35, Act: -4.342
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 40, Act: -4.342
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 59, Act: -4.342
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 58, Act: -4.342
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 50, Act: -4.342
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 34, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 55, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 47, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 5, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 84, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 43, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 63, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 60, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 78, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 94, Act: -4.345
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user

Layer 6

GATE_PROJ

Top 16 Positive Activations
Top 16 Negative Activations
Rollout 29, Act: -6.388
ĠanswerĠA.ĊĊAnotherĠanalogy:ĠifĠyouĠhaveĠaĠcontainerĠwithĠtwoĠcompartments,ĠoneĠopenĠatĠtheĠtopĠwith
Rollout 21, Act: -6.394
)Ġ-ĠyĠ=0.ĠThenĠsimilarly,ĠthatĠforcesĠf(z)Ġ=ĠzĠforĠanyĠzĠthatĠcan
Rollout 46, Act: -6.422
ĠrotatingĠframe.ĊĊAlternatively,ĠmaybeĠtheyĠrequireĠusingĠtheĠchainĠruleĠinĠsomeĠway?ĠPerhapsĠweĠneedĠtoĠtake
Rollout 93, Act: -6.432
/32âīĪ1.03ĠwhichĠmatchesĠmanual.ĠEarlierĠwhenĠtestingĠm=4:ĠsumĠvia
Rollout 71, Act: -6.434
8.ĠaĠandĠbĠasĠnegativeĠintegers.ĠPairs:Ġ(-1,-198),Ġ(-2
Rollout 93, Act: -6.436
=1.125,ĠwhichĠmatchesĠourĠmanualĠcomputationĠ(9/8).ĠSoĠtheĠformulaĠholds
Rollout 46, Act: -6.453
.ĠAlternatively,ĠmaybeĠit'sĠaskingĠtoĠapplyĠtheĠchainĠruleĠviaĠU.ĊĊAlternatively,ĠifĠtheĠenergiesĠare
Rollout 62, Act: -6.484
ĠproductĠofĠfourĠprimesĠ(repeatsĠallowed).ĊĊButĠmanualĠtrialĠmayĠbeĠneeded.ĠLet'sĠassumeĠthatĠm
Rollout 54, Act: -6.502
.ĊĊWait,Ġno:ĠLet'sĠclarify:ĊĊAcidĠstrength:ĠstrongerĠacidĠhasĠlowerĠpKa.ĠEthan
Rollout 10, Act: -6.534
ĠinĠlosingĠpositionsĠbeyondĠwhatĠIĠcalculated,ĠbutĠGivenĠmanualĠGrundyĠnumberĠcalculationĠupĠtoĠn=30
Rollout 38, Act: -6.542
ĠtoĠtheĠsoutheastĠcornerĠ(x).ĊĊ1.Ġ**TravelĠTimeĠEquations**:ĊĠĠĠ-ĠAna'sĠpath
Rollout 60, Act: -6.552
**Ċ\boxed{6}ĊĊ<|im_start|>answerĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequencyĠ\(Ġf_0Ġ=
Rollout 93, Act: -6.618
1,m=2ĠformulaĠgives1,matches.ĠManualĠcomputationĠmatches.ĠForĠm=3ĠformulaĠgives3
Rollout 62, Act: -6.626
ĠotherĠfactor).ĠNo.MaybeĠelse.ĊĊOk,ĠafterĠmanual'sĠtediousĠeffortsĠbutĠmaybeĠnoĠotherĠway,Ġto
Rollout 20, Act: -6.682
ĠperĠearlier.ĊĊButĠalternatively,ĠletĠmeĠcheckĠwithĠmanualĠcalculationĠforĠn=LCM(42,
Rollout 33, Act: -6.735
:ĊĊWithĠf(4)=13.ĊĊButĠmanualĠcountĠisĠtime-consuming.ĠHoweverĠusingĠinclusion-exclusion

UP_PROJ

Top 16 Positive Activations
Rollout 7, Act: 2.168
ĠĠ$47$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'mĠtryingĠtoĠsolveĠthis
Rollout 79, Act: 2.148
ĠfacilityĠ(8,4)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠcrosswordĠclue.
Rollout 14, Act: 2.090
ĠdxĠ\right)Ġ$$ĠĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠtoĠproveĠthisĠinequality
Rollout 93, Act: 2.073
2014}$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠproblemĠhere:
Rollout 43, Act: 2.069
0\right\rangleĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 26, Act: 2.054
mol}^{-1}$Ġ]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthisĠproblem
Rollout 41, Act: 2.053
ĠatĠmostĠtwoĠdifferentĠdigits?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠtheĠproblemĠisĠasking:ĠHow
Rollout 55, Act: 2.052
018}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠmatrixĠAĠand
Rollout 94, Act: 2.048
Ċ(D)Ġa,bĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠdetermineĠwhichĠof
Rollout 3, Act: 2.043
n)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculateĠtheĠmeanĠmolecular
Rollout 63, Act: 2.031
ĠforĠtheĠthree-variableĠcase.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠaĠfunction
Rollout 48, Act: 2.030
ĠDoesĠtheĠheaderĠcontainsĠerror?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠIPv4Ġheader
Rollout 30, Act: 2.022
3}}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'veĠgotĠthisĠproblemĠhere
Rollout 99, Act: 2.021
ĠvalueĠofĠx_3?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthis
Rollout 45, Act: 2.011
ĠofĠĠ$Ġm$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠproblemĠwhereĠa
Rollout 88, Act: 2.009
ĠchannelsĠintoĠaĠsingleĠvector)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠthe
Top 16 Negative Activations
Rollout 42, Act: -2.594
ĠandĠhexahydroĠwouldĠmeanĠit'sĠpartiallyĠhydrogenated.ĠNotĠexactlyĠsure.ĊĊ(D)Ġspiro[
Rollout 73, Act: -2.621
kĠ=Ġx(1Ġ-Ġ(n+1)x^nĠ+Ġnx^{n+1})/(1
Rollout 49, Act: -2.802
Wait,ĠbutĠinĠtheĠCantor-BendixsonĠdecomposition,ĠXĠcanĠbeĠwrittenĠasĠtheĠunionĠof
Rollout 49, Act: -2.818
sonĠtheorem?ĊĊInĠtheĠCantor-BendixsonĠtheorem,ĠweĠcanĠdecomposeĠaĠPolishĠspaceĠinto
Rollout 49, Act: -2.834
ixsonĠderivative.ĊĊTheĠCantor-BendixsonĠtheoremĠstatesĠthatĠinĠaĠseparableĠmetrizable
Rollout 49, Act: -2.862
Alternatively,ĠthinkĠofĠtheĠCantor-BendixsonĠtheorem.ĠIfĠXĠisĠaĠHausdorff
Rollout 49, Act: -2.886
ĠanĠapproachĠsimilarĠtoĠtheĠCantor-BendixsonĠtheorem?ĊĊInĠtheĠCantor-Bendixson
Rollout 49, Act: -2.913
,ĠweĠmightĠconsiderĠtheĠCantor-BendixsonĠderivative.ĊĊTheĠCantor-BendixsonĠtheorem
Rollout 49, Act: -2.929
ĠinĠsuchĠaĠspaceĠtheĠCantor-BendixsonĠderivative.ĊĊInĠsuchĠaĠspace,ĠconsiderĠX,
Rollout 49, Act: -2.967
ĠcanĠconsiderĠsomeĠkindĠofĠCantor-BendixsonĠdecomposition.ĊĊAlternatively,ĠsinceĠeveryĠsubsetĠisĠFÏĥ
Rollout 37, Act: -2.976
ĠwillĠdivideĠtheĠdeterminantĠofĠanĠ(n-1)x(n-1)ĠmatriceĠwithĠsimilarĠstructure.ĊĊ
Rollout 49, Act: -2.979
ĠbeĠbuiltĠupĠbyĠitsĠCantor-BendixsonĠsequence.ĠAtĠeachĠstage,ĠyouĠremoveĠisolatedĠpoints
Rollout 49, Act: -2.990
ĠandĠeveryĠlevelĠofĠtheĠCantor-BendixsonĠdecompositionĠisĠcountable.ĠButĠIĠamĠnotĠsure
Rollout 49, Act: -2.994
,ĠeachĠlevelĠofĠtheĠCantor-BendixsonĠdecompositionĠisĠaĠdiscreteĠsubspace.ĠForĠLindel
Rollout 49, Act: -3.043
able.ĠThenĠapplyĠtheĠCantor-BendixsonĠprocess:ĠremovingĠisolatedĠpointsĠtransfinitelyĠuntilĠweĠget
Rollout 49, Act: -3.088
ĠtopologicalĠspaces,ĠtheĠCantor-BendixsonĠprocessĠcanĠstillĠbeĠapplied.ĠLetâĢĻsĠattemptĠit

DOWN_PROJ

Top 16 Positive Activations
Rollout 0, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 1, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 6, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 3, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 4, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 7, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 14, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 9, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 15, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 12, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 10, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 11, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 8, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 16, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 17, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 20, Act: 6.269
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Top 16 Negative Activations
Rollout 42, Act: -1.492
ĠinitialĠcarbocationĠisĠatĠtheĠcentralĠcarbonĠ(tertiaryĠmaybe?),ĠperhapsĠadjacentĠtoĠaĠthree-membered
Rollout 79, Act: -1.512
ĠM,ĠA,ĠN,ĠA,ĠG,ĠE,ĠRĠ(butĠmanyĠduplicates).ĠOriginalĠlettersĠhave
Rollout 54, Act: -1.545
,ĠaĠbulkyĠnucleophileĠ(likeĠtert-butoxide)ĠisĠlessĠnucleophilicĠthanĠhydrox
Rollout 37, Act: -1.629
ant,ĠbutĠtheĠdeterminantĠisĠaĠspecificĠimmanantĠwithĠcharacters=sign.ĠButĠunlessĠthere'sĠa
Rollout 37, Act: -1.670
.ĊĊAlternatively,ĠthisĠisĠakinĠtoĠanĠimmanant,ĠbutĠtheĠdeterminantĠisĠaĠspecificĠimmanant
Rollout 62, Act: -1.732
ĠaĠproductĠofĠfourĠprimesĠisĠcalledĠaĠsphenicĠnumberĠifĠit'sĠsquare-freeĠwithĠthreeĠdistinctĠprimes
Rollout 49, Act: -1.826
ĠnotĠÏĥ-compact.ĠTherefore,ĠtheĠcontrapositiveĠwouldĠbe:ĠIfĠXĠisĠuncountableĠand
Rollout 49, Act: -1.885
,ĠcheckingĠreference:ĠInĠGeneralĠTopologyĠbyĠEngelking,ĠTheoremĠ3.8.2Ġon
Rollout 49, Act: -2.060
ĠisĠnotĠÏĥ-compact.ĠTherefore,ĠtheĠcontrapositiveĠwouldĠbe:ĠIfĠXĠisĠuncountable
Rollout 12, Act: -2.082
\nu<1$Ġ.Ġ[P.ĠTuran]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġso
Rollout 49, Act: -2.082
able.ĊĊRecallĠaĠtheoremĠfromĠ[Engelking,ĠGeneralĠTopology],Ġmaybe:ĠAĠHausd
Rollout 39, Act: -2.541
999Ġ+Ġ\cdotsĠ+Ġ\underbrace{99\ldotsĠ99}_
Rollout 2, Act: -2.555
9Ġ\timesĠ\cdotsĠ\timesĠ\underbrace{99\cdots9}_{\text
Rollout 39, Act: -2.603
999Ġ+Ġ\cdotsĠ+Ġ\underbrace{99\ldotsĠ99}_
Rollout 2, Act: -2.689
9Ġ\timesĠ\cdotsĠ\timesĠ\underbrace{99\cdots9}_{\text
Rollout 24, Act: -3.492
Ġother,ĠweĠgetĠsameĠsizes.ĠMaybeĠuseĠDilworth'sĠtheoremĠorĠsimilar.ĊĊAlternatively,ĠthinkĠinĠterms

Layer 7

GATE_PROJ

Top 16 Positive Activations
Rollout 23, Act: 6.827
5ĠBq.ĠSoĠsomeoneĠwhoĠmiscalculatesĠbutĠalsoĠuseĠlessĠapproximateĠeĠexponentĠorĠhadĠa
Rollout 98, Act: 6.610
.ĠTheĠRGBĠphaseĠisĠpartĠofĠpost-main-sequenceĠevolution.ĠSoĠifĠtheĠsystemĠisĠoldĠenoughĠto
Rollout 99, Act: 6.607
ĠifĠtheyĠsaidĠ2Ġiterations,ĠbutĠmaybeĠmistyped.ĠOr,ĠmaybeĠmyĠapproachĠisĠwrong?ĊĊWait
Rollout 43, Act: 6.564
ĠmyĠinitialĠFinalĠAnswerĠlineĠisĠperhapsĠIĠjustĠmistypedĠwhenĠIĠwroteĠ"\boxed{A}".ĠSoĠaccording
Rollout 4, Act: 6.532
0.88%,ĠorĠmaybeĠtheĠanswerĠisĠexactĠatĠ0.87%.ĊĊButĠwait,
Rollout 98, Act: 6.504
ĠMsun.ĠTheĠK3VĠisĠaĠmainĠsequenceĠstar,ĠmaybeĠaroundĠ0.7-0
Rollout 98, Act: 6.497
),ĠtheĠA0VĠisĠstillĠonĠtheĠmainĠsequence.ĠSoĠafterĠ80ĠMyr,ĠA
Rollout 26, Act: 6.495
03.4ĠisĠthreeĠsigĠfigsĠatĠmost,Ġso:ĊĊ-303.4Ġ(
Rollout 98, Act: 6.491
ĠtheĠA0VĠwouldĠstillĠbeĠonĠtheĠmainĠsequenceĠ(sinceĠitsĠlifetimeĠisĠ~370Ġmillion
Rollout 99, Act: 6.486
Ġiteration.ĠAlternatively,Ġit'sĠaĠpossibleĠmistranslation/typo,ĠbutĠbasedĠonĠtheĠexactĠsteps
Rollout 98, Act: 6.476
).ĠIfĠtheĠAĠstarĠreallyĠisĠonĠtheĠmainĠsequenceĠwithĠ0.9Msun,ĠitĠcan
Rollout 81, Act: 6.474
ĠtheĠangleĠbeingĠmeasured?ĠĊĊMaybeĠmiscalculatingĠtheĠangle.ĊĊAlternatively,ĠrecallingĠthatĠbothĠapproachesĠuse
Rollout 98, Act: 6.471
VĠ(lowerĠmass)ĠisĠstillĠonĠtheĠmainĠsequence.ĠSo,ĠtheirĠcurrentĠstateĠwouldĠhaveĠtheĠWD
Rollout 98, Act: 6.450
ĠM4VĠstarsĠtoĠstillĠbeĠonĠtheĠmainĠsequence.ĊĊ2.Ġ**WDĠ(B5ĠwhenĠMS
Rollout 98, Act: 6.439
ĠTheĠA0VĠwouldĠstillĠbeĠonĠtheĠmainĠsequenceĠbyĠtheĠtimeĠtheĠB5VĠbecomesĠaĠWD
Rollout 98, Act: 6.434
VĠ(1Msun)ĠisĠstillĠonĠmainĠsequenceĠ(lifetimeĠ~10ĠbillionĠyears).ĠTheĠM
Top 16 Negative Activations

UP_PROJ

Top 16 Positive Activations
Rollout 0, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 1, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 2, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 4, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 12, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 3, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 6, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 8, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 11, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 13, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 15, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 7, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 10, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 9, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 16, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Rollout 14, Act: 2.461
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊ
Top 16 Negative Activations
Rollout 93, Act: -2.246
(2014,1007}}ãĢĤĊĊButĠperhapsĠthere'sĠanĠidentityĠthatĠthatĠexpression
Rollout 79, Act: -2.247
ĠMags".ĠNo.ĊĊWait,ĠmaybeĠ"OrganĠChimes".ĠC,ĠH,ĠI,ĠM,
Rollout 79, Act: -2.255
ĠH,ĠO,ĠM,ĠI,ĠN,ĠGĠâĢĵĠ6Ġletters.ĠThenĠusingĠletters:ĠM
Rollout 95, Act: -2.256
åľ¨çī¹å®ļæŶåĪ»,Ġtrapezoidsä¼ļ缸éģĩ.ĊĊHowever,theĠkeyĠliesĠinĠtheĠquestionĠstatingĠeachĠregion
Rollout 95, Act: -2.264
5*2=7350,åı³.ĊĊĠSoĠSubstitute:ĊĊ=Ġ[Ġ(2*36
Rollout 95, Act: -2.273
èĢĮå¾ĹåĪ°ä¸įåIJĮçļĦæ¯ı个梯形çļĦéĿ¢ç§¯çĽ¸å¾ħ,ä½ľèĢħåı¯èĥ½è¯¯è§£äºĨéĹ®é¢ĺåı¯èĥ½åħģ许æ¯ı个åĮºåŁŁéĿ¢ç§¯
Rollout 79, Act: -2.276
ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.288
ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.290
M,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.303
,ĠS,ĠR,ĠN.ĠMaybeĠ"CarnĠSGĠEN"ĠâĢĵĠno.ĠWait,ĠlettersĠremaining
Rollout 79, Act: -2.318
ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.339
ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.356
ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.379
M,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,ĠA,ĠN,ĠG,ĠE
Rollout 79, Act: -2.490
H,ĠO,ĠM,ĠI,ĠN,ĠGĠ(homing).ĠThenĠremainingĠletters:ĠC,
Rollout 79, Act: -2.550
Ġ"minor"ĠisĠanĠanagramĠofĠ"morin"ĠbutĠthat'sĠpartĠofĠit.ĠWait,

DOWN_PROJ

Top 16 Positive Activations
Rollout 0, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 3, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 1, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 8, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 9, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 4, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 7, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 11, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 17, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 16, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 14, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 6, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 10, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 15, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 12, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 20, Act: 2.481
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Top 16 Negative Activations
Rollout 25, Act: -2.027
0Ġ\mathrm{~m}Ġ/Ġ\mathrm{s}$.ĠAlthoughĠBUĠhadĠaĠveryĠsmallĠmassĠestimated
Rollout 25, Act: -2.039
ĠâīĪĠ6.8e-5Ġm/s2.ĠEvenĠsuchĠaĠmassiveĠasteroidĠ(1e
Rollout 54, Act: -2.047
ĠThisĠalignsĠwithĠscenariosĠwhereĠtheĠrate-determiningĠstepĠ(RDS)ĠinvolvesĠaĠchargedĠreactant
Rollout 60, Act: -2.054
ĠisĠtheĠspeedĠofĠtheĠcar,Ġ2Ġm/s.ĠSoĠv_sourceĠ=Ġ2Ġm/sĊĊ
Rollout 3, Act: -2.059
Ġ\(169Ġ\,Ġ\text{m/s}\).ĊĊ**FinalĠAnswer:**Ċ\[Ċ\
Rollout 3, Act: -2.071
ĠthreeĠsigĠfigsĠjustifyĠ169Ġm/s.ĊĊThereforeĠtheĠvalidĠboxedĠanswerĠisĠ\boxed{
Rollout 22, Act: -2.072
694}Ġ\,Ġ\text{m/s}Ċ\]<|im_end|>
Rollout 22, Act: -2.097
1694Ġ\,Ġ\text{m/s}Ġ\)ĊĊ**FinalĠAnswer**:Ċ\[Ċ
Rollout 27, Act: -2.105
0^{5}Ġ\,Ġ\text{m/s}ĊĠĠĠĠĠ\]ĊĊ2.Ġ**EstimateĠthe
Rollout 60, Act: -2.113
Ġ=Ġ330Ġ\,Ġ\text{ms}^{-1}Ġ\)ĠisĠtheĠspeedĠof
Rollout 60, Act: -2.162
ĠapproachingĠtheĠsourceĠatĠaĠspeedĠofĠ2Ġm/s.ĠTheĠsoundĠreflectsĠoffĠtheĠcarĠandĠcomesĠback
Rollout 25, Act: -2.180
9300Ġ\,Ġ\text{m/s}Ġ\)ĠandĠtheĠimpactĠparameterĠ\(ĠrĠ\
Rollout 3, Act: -2.193
Ġ\boxed{169Ġ\text{Ġm/s}}.ĊĊHowever,ĠdependingĠonĠsourceĠofĠproblem,
Rollout 60, Act: -2.196
ĠairĠisĠ$330Ġ\mathrm{~ms}^{-1}$ĠandĠtheĠcarĠreflectsĠtheĠsound
Rollout 60, Act: -2.417
ĠspeedĠofĠ\(Ġ2Ġ\,Ġ\text{ms}^{-1}Ġ\).ĠWeĠneedĠtoĠfind
Rollout 60, Act: -2.554
ĠwithĠaĠspeedĠofĠ$2Ġ\mathrm{~ms}^{-1}$.ĠTheĠreflectedĠsignalĠisĠreceived

Layer 8

GATE_PROJ

Top 16 Positive Activations
Rollout 97, Act: 1.375
Ġ$\mathrm{Y}$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠhaveĠthis
Rollout 54, Act: 1.354
,ĠBĠ=Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestionĠstepĠby
Rollout 42, Act: 1.346
oct-5-eneĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠthe
Rollout 26, Act: 1.343
mol}^{-1}$Ġ]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthisĠproblem
Rollout 3, Act: 1.308
n)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculateĠtheĠmeanĠmolecular
Rollout 22, Act: 1.305
2)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠtoĠcalculateĠtheĠmean
Rollout 88, Act: 1.276
ĠchannelsĠintoĠaĠsingleĠvector)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠthe
Rollout 42, Act: 1.252
,5,6-hexahydropentaleneĠâĢĵĠPentaleneĠisĠaĠbicyclicĠcompound
Rollout 17, Act: 1.216
$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackleĠthisĠproblem
Rollout 44, Act: 1.203
mathrm{T}_{2}$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠwhich
Rollout 42, Act: 1.197
,5,6-hexahydropentaleneĊ(D)Ġspiro[3.4]
Rollout 42, Act: 1.195
4,5,6-hexahydropentaleneĊ(D)Ġspiro[3.4
Rollout 43, Act: 1.189
0\right\rangleĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 27, Act: 1.185
ĠalongĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthisĠproblemĠhere
Rollout 42, Act: 1.161
4,5,6-hexahydropentaleneĠâĢĵĠPentaleneĠisĠaĠbicyclic
Rollout 23, Act: 1.146
555ĠBqĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠstepĠby
Top 16 Negative Activations
Rollout 79, Act: -3.604
team".ĠThat'sĠnotĠinĠtheĠletters.Ġ"Unit"ĠâĢĵĠUĠnotĠpresent.Ġ"Base":ĠB
Rollout 79, Act: -3.607
ĠasĠtheĠfodder.ĠLet'sĠcount:ĠMĠIĠNĠOĠRĠCĠHĠAĠNĠGĠEĠSĠI
Rollout 79, Act: -3.651
Ġchanges"ĠandĠdefiningĠinvestigators'ĠfacilityĠisĠ"SCANNINGĠROOM."ĠScansĠ+ĠROOMĠchecksĠout)*
Rollout 79, Act: -3.686
ĠsolutionĠlikelyĠis:ĊĊ**Answer**ĠĠĊ**SCANNINGĠSHOP**ĠĠĊĊButĠhonestly,ĠtheĠcorrectĠanswer
Rollout 79, Act: -3.757
ĠTheĠproperĠanswerĠconfirmedĠbyĠcrosswordĠdatabasesĠisĠ**SCANNINGĠROOM**.ĠHowever,ĠtheĠaccurateĠanswerĠgiven
Rollout 79, Act: -3.808
ĠtheĠfinalĠanswer:ĊĊ**Answer:**ĠĠĊ**SCANNINGĠROOM**Ġ(8,4)ĠĠĊ*(
Rollout 79, Act: -3.824
)*ĊĊ**FinalĠAnswer:**ĠĠĊ**SCANNINGĠROOM**ĊĊHowever,ĠcorrectionsĠneededĠafterĠverification.ĠActual
Rollout 79, Act: -3.832
Ġ"monirchĠsaeng"ĠintoĠ"SCANNINGĠHOME"ĠâĢĶĠbutĠnotĠfitting.)ĠTheĠproper
Rollout 79, Act: -3.910
Ġletters,ĠsoĠno.ĠWait,ĠperhapsĠ"Scanning".ĠContainsĠT?ĠNo.ĠNotĠpossible.ĊĊAlternatively
Rollout 79, Act: -3.971
"ĠandĠdefiningĠinvestigators'ĠfacilityĠisĠ"SCANNINGĠROOM."ĠScansĠ+ĠROOMĠchecksĠout)*ĊĊ
Rollout 79, Act: -3.974
Ġis:*ĊĊ**Answer:**ĠĠĊ**SCANNINGĠROOM**ĊĊ*(AdmittedĠrealization:ĠAfterĠconsiderableĠtime
Rollout 79, Act: -3.978
ĠlikelyĠis:ĊĊ**Answer**ĠĠĊ**SCANNINGĠSHOP**ĠĠĊĊButĠhonestly,ĠtheĠcorrectĠanswerĠall
Rollout 79, Act: -4.001
ĠfinalĠanswer:ĊĊ**Answer:**ĠĠĊ**SCANNINGĠROOM**Ġ(8,4)ĠĠĊ*(Letters
Rollout 79, Act: -4.005
ĠUnlikely.ĠLet'sĠsee.ĠMaybeĠ"Scanning"?ĠButĠSĠisĠthereĠonce,ĠCĠisĠthere
Rollout 79, Act: -4.076
monirchĠsaeng"ĠintoĠ"SCANNINGĠHOME"ĠâĢĶĠbutĠnotĠfitting.)ĠTheĠproperĠanswer
Rollout 79, Act: -4.133
ĠproperĠanswerĠconfirmedĠbyĠcrosswordĠdatabasesĠisĠ**SCANNINGĠROOM**.ĠHowever,ĠtheĠaccurateĠanswerĠgivenĠ"

UP_PROJ

Top 16 Positive Activations
Rollout 3, Act: 3.247
n)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculateĠtheĠmeanĠmolecular
Rollout 22, Act: 3.239
2)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠtoĠcalculateĠtheĠmean
Rollout 98, Act: 3.217
Ċ(D)Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠhow
Rollout 94, Act: 3.095
Ċ(D)Ġa,bĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠdetermineĠwhichĠof
Rollout 17, Act: 3.089
$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackleĠthisĠproblem
Rollout 23, Act: 3.054
555ĠBqĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠstepĠby
Rollout 80, Act: 3.039
ĠvalueĠofĠ$N$Ġ?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 43, Act: 3.017
0\right\rangleĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 87, Act: 3.006
ĠandĠanyĠnecessaryĠnumericalĠconstants.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠtheĠelectric
Rollout 60, Act: 3.001
ĠtheĠfrequencyĠitĠhasĠreceived).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 74, Act: 2.997
,ĠandĠ$\theta$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠderiveĠanĠexpression
Rollout 82, Act: 2.996
8/69)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠanĠoptimal
Rollout 27, Act: 2.981
ĠalongĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthisĠproblemĠhere
Rollout 44, Act: 2.977
mathrm{T}_{2}$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠwhich
Rollout 88, Act: 2.966
ĠchannelsĠintoĠaĠsingleĠvector)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠthe
Rollout 46, Act: 2.955
^{2}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠtoĠfindĠtheĠsecond
Top 16 Negative Activations
Rollout 33, Act: -1.206
Ġ1Ġâī¤ĠkĠâī¤5.ĠSoĠtheĠdesiredĠcountĠisĠtotalĠpermutationsĠminusĠtheĠnumberĠofĠpermutationsĠinĠthe
Rollout 13, Act: -1.210
37789*1.03âīĪ1.34391637
Rollout 8, Act: -1.213
240=31/126âīĪ0.246.ĊĊThus,ĠtheĠanswer
Rollout 13, Act: -1.213
41492*1.03âīĪ1.55796741
Rollout 13, Act: -1.216
72322*1.03âīĪ1.51258972
Rollout 99, Act: -1.216
/1.6054934âīĪ1.1573ĊĊThus,Ġx
Rollout 90, Act: -1.220
018ĠÃĹĠ0.54ĠâīĪĠ1090.Ġ201
Rollout 31, Act: -1.229
=(1Ġ-ĠâĪļ5)/2~(approxĠ-0.618).ĊĊSoĠperhapsĠfor
Rollout 16, Act: -1.235
06Ġ-1.01587âīĪ4.0347.ĊĊWhichĠasĠa
Rollout 64, Act: -1.244
2^{2009}Ġ=8ĠâīĪĠbutĠweĠneedĠexactĠvalue.ĊĊUsingĠtheĠformulaĠfor
Rollout 99, Act: -1.255
66/1.60549âīĪ-2.1531.ĊĊThen,
Rollout 89, Act: -1.255
00/3)=200/3âīĪ66.666...ĊĊBut,Ġwhat
Rollout 13, Act: -1.256
18242*1.03âīĪâīĪ1.3047731
Rollout 70, Act: -1.265
Ġln(0.2857)ĠâīĪĠ-1.2528ĊĊln(
Rollout 69, Act: -1.268
a+b$Ġ.ĊĊĊ*2021ĠCCAĠMathĠBonanzaĠTeamĠRoundĠ#10*Ċ
Rollout 16, Act: -1.316
286Ġ+6.1935âīĪ5.0506.ĊĊ5.0

DOWN_PROJ

Top 16 Positive Activations
Rollout 15, Act: 2.349
",ĠwhichĠisĠtheĠsameĠasĠparityĠ(sumĠneigborsĠmod2Ġ=1).ĠTherefore,Ġthe
Rollout 42, Act: 2.160
ĠaĠspiroĠsystem,ĠtheĠcentralĠcarbonĠ(spiroĠatom)ĠisĠpartĠofĠbothĠrings.ĠSoĠthe
Rollout 42, Act: 2.134
lobutaneĠringsĠviaĠaĠsharedĠcarbonĠ(spiroĠcarbon),ĠthenĠmaybeĠspiro[3.4
Rollout 2, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 13, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 18, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 32, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 40, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 19, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 31, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 35, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 61, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 50, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 59, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 67, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Rollout 58, Act: 2.090
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>
Top 16 Negative Activations
Rollout 43, Act: -4.089
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠC-NOTĠgateĠwhere
Rollout 55, Act: -4.089
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĊ\[\mathbf{A
Rollout 47, Act: -4.089
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠtrainĠwithĠcross-sectionalĠareaĠ$S
Rollout 60, Act: -4.089
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequencyĠ$
Rollout 63, Act: -4.089
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$P$ĠĠdenoteĠthe
Rollout 19, Act: -4.096
$Ġ,ĠtheĠremainderĠisĠ$17$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhave
Rollout 21, Act: -4.110
}ĊĊ(b)Ġ\boxed{\text{Yes}}ĊĊ<|im_start|>answerĊ(a)ĠToĠshowĠthatĠCarlĠcanĠchoose
Rollout 63, Act: -4.114
,ĠalgebraĠA1ĠforĠtheĠthree-variableĠcase.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneed
Rollout 31, Act: -4.147
.ĠFindĠallĠpossibleĠvaluesĠofĠ$y$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtry
Rollout 45, Act: -4.170
ĠtheĠgreatestĠpossibleĠvalueĠofĠĠ$Ġm$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhave
Rollout 57, Act: -4.243
Ġ\boxed{\text{RegularĠbutĠnotĠnormal}}ĊĊ<|im_start|>answerĊToĠshowĠthatĠtheĠrequirementĠdefinesĠaĠtopology
Rollout 32, Act: -4.308
M}^{2009}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneed
Rollout 69, Act: -4.328
FinalĠAnswer**Ċ\boxed{204}ĊĊ<|im_start|>answerĊGivenĠtheĠequation:ĊĊ\[Ċ\frac
Rollout 87, Act: -4.370
(1Ġ+Ġ\sqrt{2})Ġ}.ĊĊ<|im_start|>answerĊToĠfindĠtheĠelectricĠpotentialĠatĠaĠcorner
Rollout 51, Act: -4.518
3}Ġ\sinĠ8\theta}.ĊĊ<|im_start|>answerĊToĠevaluateĠtheĠintegralĠ\(\int_
Rollout 86, Act: -4.628
{\dfrac{\pi}{16}}.ĊĊ<|im_start|>answerĊToĠfindĠtheĠminimumĠvalueĠofĠtheĠintegral

Layer 9

GATE_PROJ

Top 16 Positive Activations
Rollout 38, Act: 4.142
ĠarriveĠatĠDĠatĠtheĠsameĠtime.ĠSo,ĠperhapsĠtheĠdistancesĠtheyĠtravelĠareĠdifferent.ĠWait,Ġthat
Rollout 95, Act: 4.104
Ġk/N.ĠWhichĠexplainsĠdiscrepancyĠinĠresults.ĊĊSoĠperhapsĠtheĠparametricĠdivisionĠofĠABĠandĠACĠintoĠequal
Rollout 24, Act: 4.058
ĠinteractionĠbetweenĠbothĠrowĠdominatingĠandĠcolumnĠdominating.ĊĊSoĠperhapsĠusingĠtheĠminimalityĠinĠbothĠrowĠandĠcolumnĠdirections
Rollout 21, Act: 4.055
,ĠwhereĠSĠonlyĠcontainsĠidentityĠfunction.ĠSo,ĠperhapsĠadjustĠtheĠfunctionalĠequationĠsoĠthatĠsolutionĠmustĠbeĠfunctions
Rollout 97, Act: 4.039
ĠcontributionĠisĠstraightforward.ĊĊSoĠtheĠinitialĠassumptionĠinĠthisĠthinkĠblockĠwasĠthatĠtheYĠwouldĠbe4.6
Rollout 46, Act: 4.032
ĠandĠpositionĠareĠindependentĠvariablesĠinĠmechanics.ĠSo,ĠperhapsĠtheĠintendedĠanswerĠdoesĠnotĠincludeĠtheĠCoriolis
Rollout 56, Act: 4.025
ĠButĠtheĠproblemĠstatesĠitâĢĻsĠaĠtriangle,ĠsoĠperhapsĠA2A3A4ĠorĠA1A
Rollout 23, Act: 4.009
ĠperĠminute,ĠbutĠactivityĠisĠperĠsecond.ĠSoĠperhapsĠtheĠanswerĠisĠinĠperĠsecond,ĠsoĠlambda_alpha
Rollout 24, Act: 4.002
ĠdifferentĠminimalĠpairsĠcouldĠinterchangeĠcolumnsĠandĠrows.ĠSoĠperhapsĠthere'sĠsymmetryĠhere.ĠHowever,Ġstill,Ġhow
Rollout 56, Act: 3.993
6,ĠâĪļ3Ġ/12Ġ)ĊĊSoĠperhapsĠtheĠnewĠtriangleĠisĠA2A3A4
Rollout 33, Act: 3.983
ĠifĠn=1,ĠwouldĠnotĠapply.ĠSoĠperhapsĠnotĠuseful.ĊĊWait,ĠforĠn=2,
Rollout 7, Act: 3.982
Ġequations,ĠbutĠledĠtoĠimpossibility.ĠThus,ĠperhapsĠtheĠanswerĠisĠdifferent.ĊĊAlternatively,ĠmaybeĠN=
Rollout 24, Act: 3.981
ĠisĠconstant.ĠSimilarly,ĠsameĠforĠC.ĠSoĠperhaps,ĠsaddleĠpairsâĢĻĠrowĠsetsĠformĠaĠmatroid
Rollout 27, Act: 3.980
_Be,ĠandĠhenceĠestimateĠT_c".ĊĊSoĠperhaps,ĠstepsĠareĠfirstĠcomputeĠv_rmsĠfromĠÎĶ
Rollout 21, Act: 3.979
Ġempty.ĠButĠweĠneedĠSĠnonempty.ĠSoĠperhapsĠthatĠapproachĠisĠwrong.ĊĊAlternativeĠidea:ĠSupposeĠthat
Rollout 89, Act: 3.971
/5thĠofĠtheĠchildrenĠareĠgirls".ĊĊSoĠperhapsĠtheĠproblemĠmakerĠmixedĠupĠtheĠgenderĠinĠtheĠfractions
Top 16 Negative Activations
Rollout 24, Act: -0.464
ĠofĠallĠobjectsĠsharingĠallĠattributesĠinĠB,ĠandĠBĠisĠtheĠsetĠofĠallĠattributesĠsharedĠbyĠallĠobjects
Rollout 49, Act: -0.464
Ġ2^{2^{d(X)}},ĠwhereĠd(X)ĠisĠtheĠdensity.ĠButĠifĠourĠspace
Rollout 57, Act: -0.469
ĠĠ$Ġx$ĠĠwhoseĠdensityĠatĠĠ$Ġx$ĠĠequalsĠĠ$Ġ1$Ġ.ĠShow
Rollout 48, Act: -0.486
612Ġ4000Ġ4006Ġ6F80Ġ0A
Rollout 1, Act: -0.493
ĠĠ$Ġy\in\mathcal{H}$ĠĠsuchĠthatĠčĊ\[Ġ\left\{\frac
Rollout 1, Act: -0.527
able)ĠinĠĠ$Ġ\mathcal{H}$ĠĠsuchĠthatĠtheĠdistanceĠbetweenĠanyĠtwoĠdistinctĠpoints
Rollout 1, Act: -0.547
userĊLetĠĠ$Ġ\mathcal{H}$ĠĠbeĠanĠinfinite-dimensionalĠHilbertĠspace,Ġlet
Rollout 48, Act: -0.560
612Ġ4000Ġ4006Ġ6F80Ġ0A
Rollout 1, Act: -0.580
Ġ$ĠS$ĠĠisĠequalĠtoĠĠ$Ġd$Ġ.ĠShowĠthatĠthereĠisĠaĠpointĠĠ$Ġy
Rollout 1, Act: -0.581
>0$Ġ,ĠandĠsupposeĠthatĠĠ$ĠS$ĠĠisĠaĠsetĠofĠpointsĠ(notĠnecessarilyĠcount
Rollout 1, Act: -0.611
ĠdistanceĠbetweenĠanyĠtwoĠdistinctĠpointsĠinĠĠ$ĠS$ĠĠisĠequalĠtoĠĠ$Ġd$Ġ.ĠShow
Rollout 48, Act: -0.618
6F80Ġ0A00Ġ008BĠ5BC6ĠAEE0
Rollout 57, Act: -0.655
LetĠaĠneighborhoodĠbasisĠofĠaĠpointĠĠ$Ġx$ĠĠofĠtheĠrealĠlineĠconsistĠofĠallĠLebes
Rollout 57, Act: -0.691
ĠLebesgue-measurableĠsetsĠcontainingĠĠ$Ġx$ĠĠwhoseĠdensityĠatĠĠ$Ġx$ĠĠequals
Rollout 1, Act: -0.747
ĠHilbertĠspace,ĠletĠĠ$Ġd>0$Ġ,ĠandĠsupposeĠthatĠĠ$ĠS$ĠĠis
Rollout 57, Act: -0.836
Ġ$Ġx$ĠĠwhoseĠdensityĠatĠĠ$Ġx$ĠĠequalsĠĠ$Ġ1$Ġ.ĠShowĠthat

UP_PROJ

Top 16 Positive Activations
Rollout 93, Act: 2.105
ĠT_m(x)=Ġ±1.ĊĊButĠforĠsum_{j}[x_j^^{2m}],Ġdoes
Rollout 16, Act: 2.094
ĠtheĠsumĠasĠweĠderived,ĠwhichĠis:ĊĊSum_{k=1}^\inftyĠ[1Ġ-
Rollout 56, Act: 2.091
1/2^{n+1})=ĠSum_{k=0}^nĠ(1/2
Rollout 37, Act: 2.078
ĠisĠaĠhomogeonousĠpolynomialĠofĠdegreeĠsum_{i=1}^nĠk_i.ĠThen
Rollout 1, Act: 2.077
:ĠLetĠmeĠdefineĠyĠsuchĠthatĠyĠ=Ġsum_{i=1}^nĠa_iĠx_i
Rollout 93, Act: 2.055
Ġ+2))/(4).ĊĊInĠgeneral,forĠsum_{k=1}^mĠcos^{2m
Rollout 93, Act: 2.048
IsĠthereĠaĠgeneratingĠfunctionĠorĠidentityĠknownĠforĠsum_{k=1}^mĠcos^{2m
Rollout 37, Act: 2.040
Ġa_iĠareĠarbitraryĠintegers,ĠtheĠdeterminantĠisĠSum_{Ïĥ}Ġsgn(Ïĥ)ĠÎł_i
Rollout 64, Act: 2.033
)ĊĊĠWait,ĠstepĠbyĠstep:ĊĊForĠeachyĠ>=0Ġgiven,ĠsuchĠthatĠa_{201
Rollout 49, Act: 2.030
ĠHausdorffĠspace.ĊĊSoĠXĠ=Ġunion_{n=1}^âĪŀĠK_n,Ġwhere
Rollout 64, Act: 2.030
Ġa1=Ġx,ĠthenĠtotalĠsolutionsĠisĠSum_{Ġy=0Ġ}to8Ġ)Ġsum_(Ġx
Rollout 16, Act: 2.027
ĠofĠnĠgeometric(p)ĠvariablesĠisĠequalĠtoĠsum_{k=1}^{infty}Ġ[1
Rollout 16, Act: 2.016
Ġformula:ĊĊForĠn=1,ĠexpectationĠisĠsum_{k=1}^\inftyĠ[1Ġ-
Rollout 64, Act: 2.008
utingĠzĠ=xĠ-1,Ġexpressions:ĊĊSum_{y=0}^8Ġ(MĠ-Ky
Rollout 93, Act: 2.006
)ĠareĠrootsĠofĠaĠcertainĠpolynomial,ĠthenĠSum_{k=1}^{1007}
Rollout 93, Act: 2.003
,ĠandĠinĠourĠapproachĠresultedĠinĠtheĠformulaĠsum_{k=1}^mĠ[cos(pik
Top 16 Negative Activations
Rollout 84, Act: -0.999
ĠanswerĠisĠ\(\boxed{37}\).<|im_end|>
Rollout 44, Act: -1.005
Ġ\neqĠ\mathrm{T}_{2}$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneed
Rollout 39, Act: -1.009
FinalĠAnswer**Ċ\boxed{342}ĊĊ<|im_start|>answerĊToĠfindĠtheĠsumĠofĠtheĠdigitsĠof
Rollout 2, Act: -1.016
ĠisĠ\(\boxed{109}\).<|im_end|>
Rollout 0, Act: -1.020
FinalĠAnswer**Ċ\boxed{128}ĊĊ<|im_start|>answerĊGivenĠaĠrationalĠnumberĠbetweenĠ0Ġand
Rollout 18, Act: -1.039
(\boxed{(7,Ġ5)}\).<|im_end|>
Rollout 6, Act: -1.049
)ĠtoĠdecideĠ\(ĠxĠ\inĠLĠ\).<|im_end|>
Rollout 54, Act: -1.050
Ġ3ĠandĠ4,ĠBĠ=Ġ1<|im_end|>
Rollout 19, Act: -1.059
**FinalĠAnswer**Ċ\boxed{78}ĊĊ<|im_start|>answerĊToĠsolveĠtheĠproblem,ĠweĠneedĠto
Rollout 63, Act: -1.059
\boxed{f(p,q)Ġ=Ġpq}ĊĊ<|im_start|>answerĊToĠfindĠtheĠfunctionĠ\(Ġf(p,
Rollout 70, Act: -1.072
ĠisĠ\(\boxed{315}\).<|im_end|>
Rollout 69, Act: -1.078
FinalĠAnswer**Ċ\boxed{204}ĊĊ<|im_start|>answerĊGivenĠtheĠequation:ĊĊ\[Ċ\frac
Rollout 40, Act: -1.084
**FinalĠAnswer**Ċ\boxed{19}ĊĊ<|im_start|>answerĊTheĠparticleĠstartsĠatĠpositionĠ\((5
Rollout 92, Act: -1.108
{3.78}Ġdiopters.<|im_end|>
Rollout 41, Act: -1.111
ĠisĠ\(\boxed{927}\).<|im_end|>
Rollout 79, Act: -1.128
ANGINGĠROOMS**Ġ(8,4).<|im_end|>

DOWN_PROJ

Top 16 Positive Activations
Rollout 16, Act: 2.308
Ġ>=Ġk.ĠSinceĠeachĠTiĠisĠ>=ĠkĠwithĠprobabilityĠ(1/2)^{kĠ-Ġ1
Rollout 85, Act: 2.230
.ĠForĠexample,Ġlet'sĠsupposeĠweĠstartĠalternatelyĠlabeling.ĊĊFromĠ(0,0)=0,
Rollout 10, Act: 2.222
ĠtheĠpatternĠisĠaddingĠ2ĠthenĠ3Ġalternately.ĠHmmm.ĠIfĠthisĠcontinues,ĠthenĠthe
Rollout 16, Act: 2.198
"ĠeachĠremainingĠcouponĠ(sapling)ĠindependentlyĠwithĠprobabilityĠ1/2.ĊĊButĠisĠthereĠanĠanalogous
Rollout 63, Act: 2.170
Ġ1.ĠEachĠofĠtheseĠdirectionsĠisĠtakenĠwithĠprobabilityĠ1/2,ĠbutĠsinceĠthisĠisĠa
Rollout 33, Act: 2.163
3.ĊĊLet'sĠcomputeĠf(3).ĊĊAlternately,ĠusingĠinclusion-exclusion:ĊĊTotalĠpermutations:6
Rollout 10, Act: 2.153
Ġ**UnderstandingĠtheĠGame**:ĊĠĠĠ-ĠPlayersĠtakeĠturnsĠremovingĠsquareĠnumbersĠ(1,Ġ4,Ġ
Rollout 11, Act: 2.063
Ġzero,ĠbutĠinĠaĠnonĠsymmetricĠway.ĊĊAlternately,ĠconsiderĠallĠthreeĠvariablesĠwithĠtheĠsameĠmodulus,
Rollout 16, Act: 2.032
ĠintoĠaĠtree.ĠEachĠsaplingĠindependentlyĠsucceedsĠwithĠprobabilityĠ1/2ĠeachĠday.ĠSoĠtheĠprocess
Rollout 1, Act: 2.012
istantĠsetĠhaveĠtoĠbeĠofĠthisĠform?ĊĊAlternately,ĠletĠmeĠtryĠtoĠcreateĠy.ĠAssumeĠthat
Rollout 10, Act: 1.995
ĠopponentĠplays,ĠweĠanalyzeĠtheĠgameĠwhereĠplayersĠalternatelyĠremoveĠaĠsquareĠnumberĠofĠpebbles.ĠThe
Rollout 10, Act: 1.982
ĠonĠtheĠtable.ĠTwoĠplayersĠmakeĠtheirĠmovesĠalternately.ĠAĠmoveĠconsistsĠofĠtakingĠoffĠtheĠtableĠ
Rollout 10, Act: 1.953
ĠmeĠtryĠtoĠunderstandĠtheĠgameĠfirst.ĊĊPlayersĠtakeĠturnsĠremovingĠaĠsquareĠnumberĠofĠpebblesĠ(1
Rollout 59, Act: 1.934
ĠcoinsĠandĠaĠthirdĠcoinĠthatĠcomesĠupĠheadsĠwithĠprobabilityĠ$\frac47$Ġ.ĠJackieĠflipsĠtheĠthree
Rollout 42, Act: 1.916
ĠbondĠsuchĠthatĠthoseĠtwoĠringsĠareĠnowĠbothĠconnectedĠviaĠaĠsingleĠcarbonĠ(spiro)ĠwithĠaĠdouble
Rollout 10, Act: 1.906
ĠHmm.ĊĊAlternatively,ĠlookingĠatĠtheirĠparity:ĠalternatelyĠevenĠandĠodd.ĠButĠmaybeĠeveryĠotherĠpair.
Top 16 Negative Activations
Rollout 38, Act: -2.039
=(-x,ĠW).ĠSoĠmustĠhaveĠ-xĠ>=Ġ-L,ĠwhichĠimpliesxĠ<=L.ĠSince
Rollout 65, Act: -2.046
Ġprisms.ĊĊĠLetâĢĻsĠtryĠforĠprismĠlikeaĠ<=B=1995<=c.ĠWhen
Rollout 65, Act: -2.048
/AâĩĴA=ĠC2/B.ĊĊĠAdditionallya<=B<=CãĢĤGivenĠthat,ĠinĠthisĠcase
Rollout 7, Act: -2.050
0).ĠTheirĠnewĠscoresĠareĠoriginal+5Ġ(<=64).ĠAverageĠshouldĠbe47.ĠSo
Rollout 75, Act: -2.074
ĠcollectĠallĠtheĠsplitĠsquares,ĠwhichĠwouldĠamountĠtoĠ<=3Ġ+3Ġ+3=9Ġsquares.ĠBut
Rollout 38, Act: -2.099
ĠbecauseĠsinceĠLĠshouldĠbeĠpositive,ĠandĠxĠmustĠ<=L,ĠsinceĠotherwiseĠDĠwouldĠlieĠoutsideĠtheĠfield
Rollout 42, Act: -2.106
ĠbetweenĠtheĠtwoĠrings.ĠTheĠoct-5-eneĠpartĠsuggestsĠanĠeight-carbonĠchainĠwithĠaĠdouble
Rollout 86, Act: -2.115
/2],ĠxĠ>=ĠsinxĠsinceĠsinxĠ<=xĠforĠxĠ>=0,ĠandĠcosxĠ<=
Rollout 42, Act: -2.125
iro[3.4]oct-5-eneĠwouldĠhaveĠaĠspiroĠunionĠwhereĠoneĠringĠis
Rollout 25, Act: -2.146
541ĠkmĠaltitude.ĊĊNeedĠTotalĠÎĶxĠ>=Ġ3541Ġkm.ĊĊThus,ĊĊ(
Rollout 42, Act: -2.163
iro[3.4]oct-5-eneĠâĢĵĠAĠspiroĠcompoundĠwithĠtwoĠrings,Ġone
Rollout 38, Act: -2.207
.ĠSoĠxĠcannotĠexceedĠL.ĠHence,ĊĊxĠ<=ĠLĊĊGivenĠL=155/1
Rollout 42, Act: -2.217
'-bi(cyclobutan)]-1-eneĊ(B)Ġ1,2-dimethyl
Rollout 42, Act: -2.259
iro[3.4]oct-5-eneĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġso
Rollout 38, Act: -2.313
ĠhaveĠ-xĠ>=Ġ-L,ĠwhichĠimpliesxĠ<=L.ĠSinceĠL=155/1
Rollout 86, Act: -2.330
ĠxĠwithĠsinx.ĊĊWait,ĠbutĠ0Ġ<=xĠ<=pi/2.ĠHere,Ġsinx

Layer 10

GATE_PROJ

Top 16 Positive Activations
Rollout 54, Act: 3.360
ĠWait,ĠmaybeĠI'mĠconfusingĠthings.ĠLet'sĠrecallĠthatĠinĠpolarĠaproticĠsolvents,Ġnucle
Rollout 79, Act: 3.330
Ġ3.ĠNo.ĊĊHmm.ĊĊWait,ĠIĠrecallĠhearingĠtheĠtermĠ"SceneĠofĠCrime"ĠâĢĵĠthat
Rollout 22, Act: 3.295
ĠK.ĠButĠmaybeĠcheckĠboth.ĊĊButĠletĠmeĠrecall.ĠIfĠIĠlookĠupĠtheĠformulaĠforĠmeanĠspeed
Rollout 23, Act: 3.258
ButĠmaybeĠtheĠpreciseĠcalculationĠisĠdifferent.ĠLetĠmeĠrecallĠthatĠe^-70=ĠapproxĠĠapproxĠ
Rollout 69, Act: 3.247
3).ĠPerhapsĠusingĠmultiple-angleĠidentities.ĊĊLetĠmeĠrecallĠtheĠidentity:Ġsin(3Ġtheta)Ġ=Ġ
Rollout 73, Act: 3.245
ĠcodeĠhere,ĠmaybeĠsimulate.ĊĊAlternatively,ĠifĠIĠwriteĠdownĠexponentsĠandĠcomputeĠterms.ĊĊAlternatively,Ġmaybe
Rollout 12, Act: 3.237
ĠofĠpowerĠfunctions.ĠButĠnotĠsure.ĠLetĠmeĠrecallĠtheĠBetaĠfunction:ĊĊB(p,Ġq)Ġ=
Rollout 27, Act: 3.236
ĠformulaĠforĠT_cĠinĠBEC.ĊĊLetĠmeĠrecallĠtheĠformulaĠforĠT_c.ĠForĠBose-Einstein
Rollout 86, Act: 3.222
ĠtoĠâĪ«x2Ġcosx.ĠLetĠmeĠrecall.ĊĊWait,Ġlet'sĠcomputeĠâĪ«x2
Rollout 33, Act: 3.211
=3:Ġ-,Ġetc.ĠWait,ĠletĠmeĠrecall.ĊĊInĠinclusion-exclusion,ĠtheĠgeneralĠtermĠfor
Rollout 86, Act: 3.205
16.ĠThereforeĠminimum.ĊĊButĠwaitĠletĠmeĠrecallĠotherĠmethods.ĠPerhapsĠapplyĠinequality.ĊĊForĠexample,
Rollout 86, Act: 3.200
ĠâĪ«xĠcos2xĠdx?ĊĊLetĠmeĠrecall.ĠNo,ĠbutĠintegrateĠâĪ«xĠcos2
Rollout 15, Act: 3.200
entĠoverĠGF(2).ĊĊNow,ĠletĠmeĠrecall:ĠforĠtheĠadjacencyĠmatrixĠofĠaĠpathĠgraph,
Rollout 45, Act: 3.193
(0)ĠusingĠtheĠrecurrenceĠrelation.ĠLetĠmeĠrecallĠtheĠrecurrence.ĠEarlier,ĠwhenĠweĠsetĠxĠ+
Rollout 75, Act: 3.184
ĠisĠ32Ġpossible?ĠWait,Ġlet'sĠrecallĠthatĠ32ĠcanĠalsoĠbeĠwrittenĠasĠsums
Rollout 79, Act: 3.183
ĠisĠtheĠanswerĠhere...ĊĊWait,ĠIĠthinkĠIĠrecallĠthatĠsomeĠcrosswordsĠuseĠtermsĠlikeĠ"SafeĠHouse
Top 16 Negative Activations
Rollout 1, Act: -0.635
x'||2Ġ=Ġ0ĠâĩĴĠ⣨xĠ-Ġx',Ġt⣩Ġ=Ġ||x||2Ġ-Ġ||
Rollout 54, Act: -0.636
ĠsoĠethanethiolateĠwouldĠbeĠCH3CH2SâĪĴ.ĠOkay,ĠsoĠallĠnucleoph
Rollout 1, Act: -0.643
ĠS'ĠisĠ||z||Ġ=Ġ||xĠ-Ġx0||,ĠbutĠinĠtheĠoriginalĠsetĠS,
Rollout 46, Act: -0.665
Ġfollows:ĊĊÏģ1Ġ=Ġsqrt(Ġ(xĠ-Ġx1)^2Ġ+Ġ(yĠ-Ġy1)^
Rollout 1, Act: -0.677
H}$ĠĠsuchĠthatĠtheĠdistanceĠbetweenĠanyĠtwoĠdistinctĠpointsĠinĠĠ$ĠS$ĠĠisĠequalĠto
Rollout 46, Act: -0.703
Ġ/ĠÏģ12Ġ*Ġ(Ġ(xĠ-Ġx1)/Ïģ1,Ġ(yĠ-Ġy1
Rollout 1, Act: -0.717
Ġ=Ġx0Ġ+Ġz'ĠisĠ||zĠ-Ġz'||Ġ=Ġd.ĠNow,ĠweĠneedĠtoĠfind
Rollout 54, Act: -0.740
ethiolate.ĠEthanethiolĠisĠCH3CH2SH,ĠsoĠethanethiolateĠwould
Rollout 46, Act: -0.762
ĠthenĠÏģ1Ġ=Ġsqrt(Ġ(xĠ-Ġx1)^2Ġ+Ġ(yĠ-Ġy1)^
Rollout 54, Act: -0.774
,ĠethanethiolateĠwouldĠbeĠCH3CH2SâĪĴ)ĊĊWait,ĠtheĠnames:Ġ
Rollout 54, Act: -0.778
anethiolateĠ(HSâĪĴĠorĠCH2CH2SâĪĴ),ĠisĠaĠweakerĠbaseĠthanĠhydro
Rollout 46, Act: -0.789
ĠthenĠÏģ1Ġ=Ġsqrt(Ġ(xĠ-Ġx1)^2Ġ+Ġ(yĠ-Ġy1)^
Rollout 54, Act: -0.795
ĠresonanceĠstabilization?ĊĊWait,ĠpropionateĠ(CH2CH2COOâĪĴ)ĠisĠaĠresonance-stabil
Rollout 54, Act: -0.840
.88,ĠsoĠpropionateĠ(CH2CH2COOâĪĴ)ĠisĠaĠweakerĠbaseĠthan
Rollout 54, Act: -0.868
:ĊĊ5.ĠEthanethiolateĠ(CH3CH2SâĪĴ):ĠmostĠpolarizable,ĠweakerĠbase
Rollout 54, Act: -0.955
OHâĪĴ)Ċ3.ĠPropionateĠ(CH2CH2COOâĪĴ)Ċ4.ĠMethanolĠ(

UP_PROJ

Top 16 Positive Activations
Rollout 65, Act: 1.944
ĠbeĠcutĠsoĠtoĠgiveĠaĠsimilar.ĊĊBut,ĠsupposeĠtheĠprismĠP(a,1995,c
Rollout 65, Act: 1.917
Ġ2:ĠRatioĠkĠifĠpermute.ĊĊSuppose:ĊĊaĠ/1995Ġ=19
Rollout 65, Act: 1.908
ĠR.ĠIfĠratiosĠnewĠprismĠorderĠdifferent.ĊĊSupposeĠtheĠnewĠprismĠisĠconsideredĠordered:ĠeitherĠaĠ<=
Rollout 21, Act: 1.878
ĠSoĠceil(x)ĠisĠallowed.ĊĊAnotherĠexample,ĠsupposeĠweĠdefineĠf(x)=Ġfloor(x)ĠifĠx
Rollout 62, Act: 1.851
ĠbutĠthenĠtheĠmultiplicityĠrequiresĠtheĠpower.ĊĊSupposeĠforĠ11*11*11
Rollout 65, Act: 1.851
ĠscaledĠversion.ĠAgainĠtheĠsameĠimpasse.ĊĊSupposeĠsimilarityĠrequiresĠthatĠtheĠnewĠproportionsĠmustĠequalĠtheĠold
Rollout 28, Act: 1.837
ĠthereĠaĠbetterĠwayĠtoĠexpressĠit?ĊĊAlternatively,Ġsuppose,ĠsetĠparametersĠorĠexpressĠinĠtermsĠofĠcentroid's
Rollout 65, Act: 1.834
Ġ1:ĠdimensionsĠareĠscaledĠbyĠk.ĊĊSupposeĠtheĠsimilarĠprismĠhasĠsidesĠka,Ġkb,Ġkc
Rollout 65, Act: 1.826
ĠtoĠnoĠsolutions.ĊĊAlternatively,ĠanotherĠpermutation?ĊĊSupposeĠafterĠcuttingĠc1,ĠthenĠnewĠprism'sĠratios
Rollout 93, Act: 1.826
m,m))/4^{m}.ĊĊNow,ĠsupposeĠweĠconsiderĠtheĠcentralĠtermĠC(2m,m
Rollout 27, Act: 1.818
Ġ~1.49e5Ġm/s.ĠWhichĠconfirmsĠitsĠtheĠsameĠresult.ĊĊHence,Ġgiven
Rollout 65, Act: 1.813
Ġk*c].ĊĊPossiblyĠre-ordered:ĊĊSupposeĠoriginalĠorderedĠ[a,1995,
Rollout 11, Act: 1.807
ÏĢ.ĠForcingĠabc=1.ĊĊHowever,ĠsupposeĠarrangingĠaĠ=kĠe^{iθ},Ġb
Rollout 1, Act: 1.787
)/2.ĊĊPuttingĠallĠthisĠtogether.ĊĊNow,ĠsupposeĠthatĠweĠcanĠmodelĠsomeĠkindĠofĠsystem.ĊĊLet
Rollout 65, Act: 1.782
1995,1995).ĠWhichĠisĠsimilarĠtoĠtheĠoriginalĠprismĠ(a,1
Rollout 14, Act: 1.782
ĠtheĠfunctionĠsuchĠthatĠmultipleĠvariablesĠareĠinvolved.ĊĊSupposeĠIĠmakeĠaĠsubstitutionĠx=ĠcyĠwhereĠcĠis
Top 16 Negative Activations
Rollout 49, Act: -1.763
dorffĠspaceĠĠ$X$ĠĠisĠĠ$\sigma$Ġ-compact,ĠthenĠĠ$X
Rollout 45, Act: -1.765
:ĠZĠ\toĠZ$ĠĠbeĠsuchĠthatĠĠ$Ġf(1)Ġ\equal{}Ġ1
Rollout 96, Act: -1.780
ĠLet'sĠcomputeĠthatĠfirst.ĊĊLineĠLĠhasĠslopeĠ5/12ĠandĠpassesĠthroughĠpointĠA(
Rollout 76, Act: -1.781
ĠthatĠhaveĠatĠleastĠoneĠ2x2Ġall-redĠsquare,ĠandĠthenĠsubtractĠthatĠfromĠtheĠtotalĠto
Rollout 57, Act: -1.792
ĠsetsĠcontainingĠaĠpointĠ\(ĠxĠ\)ĠwithĠdensityĠ1ĠatĠ\(ĠxĠ\).ĊĊ###ĠRegularityĊĊ
Rollout 33, Act: -1.795
ĠneedĠtoĠfindĠtheĠnumberĠofĠpermutationsĠofĠtheĠnumbersĠ1ĠthroughĠ6ĠsuchĠthatĠforĠeachĠkĠfrom
Rollout 56, Act: -1.796
ĠstartingĠwithĠaĠright-angledĠtriangle,ĠwithĠanglesĠ90°,Ġ60°,Ġand
Rollout 45, Act: -1.796
{}4m$ĠĠforĠallĠnon-negativeĠintegersĠĠ$Ġx$Ġ,ĠfindĠtheĠgreatestĠpossibleĠvalueĠof
Rollout 5, Act: -1.798
zoidĠintoĠtwoĠregionsĠwithĠareasĠinĠtheĠratioĠ2:3.ĠThen,ĠweĠneedĠtoĠfind
Rollout 81, Act: -1.801
A\)ĠdividingĠ\(BC\)ĠintoĠsegmentsĠofĠlengthĠ3ĠandĠ17,ĠweĠneedĠtoĠfind
Rollout 28, Act: -1.825
Ġ$G$ĠĠbeĠtheĠcentroidĠofĠtheĠtriangleĠĠ$OAB.$ĠĠ$(a)$ĠĠPro
Rollout 63, Act: -1.837
P$ĠĠdenoteĠtheĠsetĠofĠallĠorderedĠpairsĠĠ$Ġ\left(p,q\right)$ĠĠof
Rollout 15, Act: -1.873
ĠĠlampsĠareĠplacedĠinĠaĠline.ĠAtĠminuteĠ0,ĠsomeĠlampsĠareĠonĠ(maybeĠallĠof
Rollout 11, Act: -1.889
Ġ$c$ĠĠareĠthreeĠcomplexĠnumbersĠwithĠproductĠĠ$1$Ġ.ĠAssumeĠthatĠnoneĠofĠĠ$
Rollout 14, Act: -1.902
0,\infty)$ĠĠisĠintegrableĠandĠĠ$$Ġ\int_0^\inftyĠf(x
Rollout 72, Act: -1.983
Ġ(notĠnecessarilyĠdistinct)ĠrootsĠofĠunityĠofĠorderĠĠ$ĠnĠ$ĠĠwhoseĠsumĠisĠĠ$ĠĊ

DOWN_PROJ

Top 16 Positive Activations
Rollout 11, Act: 2.277
Ġ\overline{z}ĠisĠrealĠ(theĠmodulusĠsquared).ĠSinceĠpĠisĠaĠcombinationĠofĠa,
Rollout 11, Act: 2.175
ĠSoĠtheyĠcanâĢĻtĠbeĠreal,ĠevenĠifĠtheirĠmodulusĠisĠnotĠ1.ĠSoĠallĠofĠa,
Rollout 11, Act: 2.029
Ġc=Ġe^{-i2θ}.ĠButĠmodulusĠofĠaĠandĠbĠisĠ1,ĠwhichĠisĠprohibited
Rollout 87, Act: 1.992
795ĊĊOtherĠcombinations.ĠForĠinstance,ĠconsideringĠhypergeometricĠfunctionsĠorĠsimilar,ĠthereĠisĠno
Rollout 84, Act: 1.968
.e.,Ġ\(Ġ\frac{1}{2}Ġ\)Ġunits)ĠandĠdrankĠtwiceĠasĠmuchĠ(i
Rollout 28, Act: 1.961
ĠsymmetricĠstructureĠinĠaĠandĠb.ĠSo,ĠmaybeĠconsideringĠdividingĠtermsĠappropriately.ĊĊAlternatively,ĠsupposeĠletâĢĻsĠset
Rollout 49, Act: 1.954
ableĠHausdorffĠspace,ĠthenĠthereĠexistsĠsomeĠsubspaceĠofĠXĠthatĠisĠnotĠÏĥ-compact
Rollout 1, Act: 1.934
ĠtheĠdistanceĠbetweenĠanyĠtwoĠisĠtheĠsame,ĠtheyĠformĠaĠregularĠsimplex.ĠTheĠregularĠsimplexĠhasĠtheĠproperty
Rollout 18, Act: 1.932
ĠgeneratedĠbyĠtwoĠelementsĠ(byĠtheĠstructureĠtheoremĠforĠfinitelyĠgeneratedĠabelianĠgroups),ĠorĠevenĠa
Rollout 76, Act: 1.914
)ĠorĠoverlappingĠinĠaĠsingleĠsquare.ĊĊTherefore,ĠconsideringĠallĠpairsĠofĠ2x2Ġsquares,Ġlet
Rollout 11, Act: 1.909
Ġ1.ĠLetĠkĠbeĠaĠcomplexĠnumberĠwithĠmodulusĠnotĠ1,Ġe.g.,ĠkĠ=Ġ
Rollout 81, Act: 1.905
Ġtrust?ĠLet'sĠnote:ĊĊInĠcoordinateĠsystem,ĠconsideringĠangleĠatAĠasĠangleĠbetweenĠAB(h/3
Rollout 1, Act: 1.875
ĠButĠactuallyĠno,ĠinĠtheĠprevious,ĠifĠweĠtakeĠmultipleĠpoints,ĠweĠneedĠtoĠensureĠthatĠtheyĠlie
Rollout 37, Act: 1.871
ĠsawĠearlierĠisn'tĠtheĠcase.ĠForĠexample,ĠtakingĠallĠa_iĠdistinctĠprimes.ĊĊWait,ĠperhapsĠwe
Rollout 27, Act: 1.871
Ġnuclei,Ġ\(ĠV_{\text{Be}}Ġ\),ĠandĠestimateĠtheĠcriticalĠtemperatureĠ\(ĠT_{\
Rollout 60, Act: 1.869
ĠreceivesĠtheĠsound.ĠTheĠobservedĠfrequencyĠ\(Ġf'Ġ\)ĠisĠgivenĠbyĠtheĠDopplerĠeffectĠformulaĠfor
Top 16 Negative Activations
Rollout 12, Act: -1.883
).ĠStartĠwith:ĊĊf(ν)Ġ=ĠâĪ«_{ν}^1Ġ[Ġ(t2Ġ-
Rollout 51, Act: -1.887
sinx|ĠdxĠ=Ġ2ĠâĪ«0^{ÏĢ}Ġcos8xĠsinxĠdxĠ=Ġ
Rollout 17, Act: -1.888
r/a0}Ġ(1/r3)ĠâĪ«0^ÏĢĠ(3Ġcos2θĠ-Ġ
Rollout 51, Act: -1.895
Ġappropriately.ĠSoĠtheĠintegralĠbecomes:ĊĊIntegralĠ=ĠâĪ«0^θĠsin(8x)*(-sin
Rollout 12, Act: -1.895
Ġto1.ĠSoĠf(ν)Ġ=ĠâĪ«_ν^1Ġ[1Ġ/Ġsqrt(Ġ(
Rollout 51, Act: -1.915
Ġsin8tĠ(-sint)ĠdtĠ=ĠâĪ«0^{ÏĢ}Ġsin8tĠsintĠdtĊĊ
Rollout 12, Act: -1.926
Ġis:ĊĊF(ÏĨ,Ġk)Ġ=ĠâĪ«_0^phiĠ[Ġ1Ġ/Ġsqrt(
Rollout 26, Act: -1.931
0083Ġ*Ġ298ĠâīĪĠ0.0083Ġ*Ġ3
Rollout 9, Act: -1.933
_upperĠandĠy_lowerĠfromĠaĠtoĠbĠisĠâĪ«[aĠtoĠb]Ġ(y_upperĠ-Ġy
Rollout 12, Act: -1.953
Ġis:ĊĊF(ÏĨ,Ġk)Ġ=ĠâĪ«_0^ÏĨĠ[Ġ1Ġ/Ġsqrt(
Rollout 17, Act: -1.973
u2Ġ-1)Ġ(-du)Ġ=ĠâĪ«_{-1}^1Ġ(3u2
Rollout 86, Act: -1.975
Ġpi/2.ĊĊSoĠnowĠcomputeĠ3ĠâĪ«x2ĠcosĠxĠdx.ĠIntegrateĠbyĠparts
Rollout 17, Act: -1.976
,ĠtheĠintegralĠbecomes:ĊĊ**I**Ġ=ĠâĪ«Ġ[Ġ(μ0Ġ/Ġ4ÏĢ)Ġ(
Rollout 38, Act: -1.993
/6.2)(WĠ+Ġx)ĠâīĪĠ(0.3871)(WĠ+
Rollout 87, Act: -2.017
1/2)Ġr2ĠlnĠrĠ-ĠâĪ«Ġ(1/2)ĠrĠdrĠ=Ġ(1
Rollout 87, Act: -2.105
Ġ=Ġ(1/(4ÏĢε0))ĠâĪ«Ġ(ÏģĠ/Ġr)ĠdV,ĠwhereĠÏ

Layer 11

GATE_PROJ

Top 16 Positive Activations
Rollout 90, Act: 4.052
ĠitselfĠisĠ4Ġdigits).ĠWait,ĠletĠmeĠthinkĠagain.ĊĊWait,ĠholdĠon.Ġ20
Rollout 89, Act: 4.052
:2ĠboysĠtoĠgirls?ĠWait,Ġlet'sĠthink.ĠIfĠtheĠratioĠofĠgirlsĠtoĠtotalĠisĠ
Rollout 91, Act: 4.047
ĠBDĠisĠ45.ĠHmm.ĠLet'sĠthink.ĊĊButĠperhapsĠit'sĠbetterĠtoĠassignĠcoordinates.
Rollout 96, Act: 4.037
ĠareĠfoundĠbyĠperpendicularĠprojections.ĠWait,ĠletĠmeĠthink.ĊĊSupposeĠinĠtheĠnewĠcoordinateĠsystem:ĊĊ-
Rollout 95, Act: 3.981
ĠisĠaddedĠincrementally.ĠWait,ĠmaybeĠlet'sĠthinkĠofĠtheĠareaĠofĠtriangleĠAPkQkĠis
Rollout 36, Act: 3.978
Ġlet'sĠalsoĠconsiderĠEquationĠ(4).ĠLetĠmeĠthink.ĠLet'sĠseeĠEquationĠ(4):ĊĊx2
Rollout 28, Act: 3.966
2b2/a2.ĊĊHmm.ĠLetĠmeĠthinkĠofĠthisĠasĠaĠquadraticĠformĠinĠvariablesĠAĠand
Rollout 20, Act: 3.959
ĠbutĠneedĠtoĠcheckĠcarefully.ĠWait,Ġlet'sĠthink:ĊĊButĠforĠmodulusĠ43:ĠnĠâī
Rollout 24, Act: 3.958
ĠmightĠgetĠtooĠcomplicated.ĊĊAlternativeĠapproach.ĠLet'sĠthinkĠaboutĠifĠweĠcanĠshowĠ|R1|Ġ=
Rollout 79, Act: 3.942
.ĠButĠsometimes,ĠpartsĠcanĠoverlap.ĠLetĠmeĠthinkĠagain.ĊĊIfĠ"Noticed"ĠisĠtheĠan
Rollout 8, Act: 3.935
ĠwantĠtoĠcross-verify.ĊĊWait,ĠletâĢĻsĠthinkĠofĠanotherĠapproach.ĠForĠinstance,ĠimagineĠthatĠwe
Rollout 36, Act: 3.927
ĠcallĠthisĠequationĠ(d)ĊĊButĠlet'sĠalsoĠthinkĠaboutĠequationĠ(1).ĠFromĠequationĠ(1):ĊĊ
Rollout 54, Act: 3.923
ophilic?)ĊĊButĠthisĠisĠconfusing.ĠLet'sĠthinkĠinĠtermsĠofĠspecificĠexamples.ĊĊInĠaqueousĠsolution
Rollout 29, Act: 3.912
ĠlevelsĠareĠdifferentĠbecauseĠofĠpressureĠdifferences.ĠLet'sĠthinkĠofĠitĠasĠaĠU-tube.ĠAĠU
Rollout 9, Act: 3.907
ĠtoĠEM,ĠhĠcanĠbeĠanything.ĠLet'sĠthink.ĠIfĠhĠisĠlessĠthanĠ10,
Rollout 85, Act: 3.905
ĠdeterminedĠinĠaĠdifferentĠway.ĊĊWait,Ġlet'sĠthinkĠofĠtheĠproblemĠasĠsimilarĠtoĠaĠgridĠwhereĠeach
Top 16 Negative Activations
Rollout 95, Act: -2.254
3675Ġ)=Ġ(15Ġsqrt(3))/(2*3675Ġ)=
Rollout 97, Act: -2.256
kĠt}ĊĊThus,ĠsolvingĠyieldsĠt=Ġln(10}/(2k))=2.
Rollout 97, Act: -2.260
ĠYĠisĠ2.30,ĠsinceĠln(10)/Ġ(2Ġk}=Ġln(1
Rollout 87, Act: -2.263
Ġε0)Ġ[ĠÏĢ/3Ġ-Ġ(sqrt(3)Ġ-Ġ1)/2Ġ]ĊĊButĠI
Rollout 14, Act: -2.275
Ġ+Ġa/2.ĊĊNeedĠtoĠshowĠ(Ġsqrt(2ĠKĠaĠ)Ġ+aĠ/2Ġ)3
Rollout 14, Act: -2.287
ĠcubeĠweĠhave:ĊĊI^3Ġ<=Ġ(sqrt(ĠJ)Ġsqrt(AĠ)Ġ+ĠK/AĠ)^
Rollout 95, Act: -2.290
0ãĢĤĊĊĠToĠhaveĠthisĠlengthĠrational,Ġ(sqrt(3)Ġk)/490ĠmustĠequalsĠrational
Rollout 87, Act: -2.295
)/(4ĠÏĢĠε0)Ġ*Ġ(3Ġln(1Ġ+ĠâĪļ2)Ġ+ĠâĪļ3
Rollout 87, Act: -2.337
ĠbyĠstep.ĠTherefore,ĠexpressionĠis:ĊĊ3Ġln(1Ġ+Ġsqrt(2))Ġ+Ġsqrt(3
Rollout 56, Act: -2.342
A2|=Ġ(1/14)(sqrt(559)/12)=Ġsqrt(5
Rollout 14, Act: -2.343
ĠaĠ)(a/2Ġ)Ġ+3Ġ(sqrt(2ĠKĠaĠ))(a2/4Ġ)
Rollout 95, Act: -2.360
Ġ)Ġ)/3675=Ġ(sqrt(675)/2)/3675
Rollout 97, Act: -2.373
ĠifĠrounded,)ĠbutĠlikelyĠtheĠpreciseĠanswerĠisln(10}/(2kâīĪ5ÃĹ1
Rollout 97, Act: -2.387
=1.45âĨĴĠt=Ġ(Ġln(10)/Ġ(2Ġk))=23
Rollout 97, Act: -2.405
.45}ĠleadsĠtoĠt=Ġ(ln(10))/(2ÃĹ5e-4)}
Rollout 95, Act: -2.471
Ġ)ãĢĤĊĊĠThusĠlengthĠofĠeachĠsegmentĠisĠ(sqrt(3)Ġk)/490ãĢĤĊĊĠToĠhave

UP_PROJ

Top 16 Positive Activations
Rollout 14, Act: 2.537
Ġ/Ġk^3.ĠĠSoĠit'sĠevenĠthenĠnotĠtight.ĠTherefore,ĠmaybeĠtheĠ8Ġhere
Rollout 62, Act: 2.488
.ĠNotĠthatĠeasyĠtoĠdoĠmanually.ĊĊAlternatively,ĠthinkĠofĠm=55:ĊĊm=55
Rollout 95, Act: 2.438
50.ĊĊĠButĠhereĠisĠtheĠdistance.ĠSoĠthen,ĠareaĠofĠtrapezoidĠwouldĠbeĠaverage
Rollout 41, Act: 2.424
ĠaĠa,Ġetc.ĠThusĠ6.ĠSoĠthenĠforĠeachĠpair,Ġ6Ġnumbers.ĠSoĠ
Rollout 79, Act: 2.423
ĠA,ĠG,ĠS).ĊĊSoĠthat'sĠalsoĠnotĠpossible.ĊĊHmm.ĠAfterĠallĠthisĠtime,ĠI
Rollout 75, Act: 2.413
ĠhereĠ8*4=32.ĠSoĠthenĠneedĠtoĠexpress32ĠasĠsumĠofĠthreeĠeven
Rollout 75, Act: 2.386
Ġ1+1+1+1.ĠSoĠthenĠ49Ġ+4Ġ(remaining)+4Ġ+
Rollout 62, Act: 2.381
1)ĠshouldĠbeĠproductĠofĠtwoĠprimes.ĠSoĠthenĠ11ĠmultipliedĠbyĠtheseĠtwoĠprimesĠwouldĠmake
Rollout 75, Act: 2.376
4Ġ+4Ġ+0=ĠallowingĠzero.ĠSoĠthenĠscalingĠback,Ġsqrt(4ÃĹ4)=4
Rollout 51, Act: 2.376
/63.ĠThatĠpartĠchecksĠout.ĠSoĠthenĠtheĠanswerĠisĠ-4/63Ġsin8
Rollout 96, Act: 2.375
ĠindicatingĠcalculationĠmessedĠsignĠorĠoriginalĠparameters.ĊĊWait,ĠthinkingĠback:ĠWhenĠweĠsolveĠtheĠvectorĠequation:ĠOP
Rollout 75, Act: 2.364
62+32+22.ĠSoĠthenĠtotalĠthreeĠsquares.ĠButĠifĠIĠneedĠthreeĠsquares
Rollout 23, Act: 2.363
Ġactivity,ĠnotĠTotalĠactivityĠorĠbetaĠdecay.ĊĊSoĠthenĠwhyĠisĠDĠcloseĠtoĠtheĠtotalĠactivity.ĊĊAlternatively
Rollout 11, Act: 2.350
EvenĠthisĠdirectionĠisĠnotĠleadingĠanywhere.ĠFinally,ĠthinkĠofĠtheĠanswer.ĠSinceĠinĠolympiadĠproblems
Rollout 75, Act: 2.348
2Ġ+Ġ(2a4)^2.ĠSoĠthen,ĠtotalĠ8MĠ+3Ġ=ĠsumĠof
Rollout 75, Act: 2.339
Ġ4Ġ+4+1=9.ĠSoĠthen,Ġ25Ġ+Ġ(4Ġ+4Ġ+
Top 16 Negative Activations
Rollout 87, Act: -1.505
ĠmeĠletĠuĠ=Ġln(1Ġ+Ġsqrt(1Ġ+Ġr2)),ĠdvĠ=ĠrĠdrĊĊThen
Rollout 51, Act: -1.512
Ġâī¤ĠθĠâī¤ĠÏĢ,ĠθĠ+ĠÏĢĠâī¤Ġ2ÏĢĠwhenĠθĠâī¤ĠÏĢ.ĠWait
Rollout 14, Act: -1.513
ityĠtoĠshow:ĊĊ\[ĊI^3Ġ\leqĠ8JK.Ċ\]ĊĊThus,Ġthe
Rollout 87, Act: -1.517
«0^1Ġln[(1Ġ+Ġsqrt(1Ġ+Ġη2Ġ+ĠÎ2))/sqrt
Rollout 14, Act: -1.533
ftyĠf(x)ĠdxĠ\right)^3Ġ\leqĠ8\left(\int_0^\
Rollout 18, Act: -1.539
ĠHĠisĠtheĠkernelĠofĠtheĠhomomorphismĠÏĨ:ĠZ2ĠâĨĴĠZ/xZĠdefinedĠbyĠÏĨ(k
Rollout 14, Act: -1.541
ĠthenĠourĠgoalĠreducesĠtoĠshowingĠthatĠI^3Ġâī¤8,ĠwhichĠisĠequivalentĠtoĠIâī¤2.
Rollout 64, Act: -1.552
,ĠtheĠinequalityĠa_{2010}Ġâī¤Ġ2^{2012}Ġbecomes:ĊĊ
Rollout 14, Act: -1.579
ĠinequalityĠ$(aĠ+ĠbĠ+Ġc)^3Ġ\leqĠ...$.ĠHmm,ĠbutĠnotĠdirectlyĠapplicableĠhere
Rollout 14, Act: -1.593
1.ĠButĠthenĠtheĠinequalityĠrequiresĠI^3Ġâī¤8JĠK,ĠwhichĠwouldĠgiveĠ8Ġas
Rollout 14, Act: -1.596
ĠbutĠtheĠquestionĠstatesĠtoĠproveĠthatĠI^3Ġâī¤8ĠJĠK.ĠHere,ĠifĠIâīĪ1
Rollout 14, Act: -1.614
ĠWeĠneedĠtoĠshowĠthatĠ\(ĠI^3Ġ\leqĠ8JKĠ\).ĊĊWeĠsplitĠtheĠintegral
Rollout 14, Act: -1.614
=t.ĠThenĠtheĠinequalityĠstatesĠthatĠI^3Ġâī¤8Ġ*1Ġ*t.ĊĊToĠmakeĠthisĠwork
Rollout 14, Act: -1.627
1.ĠThenĠneedĠtoĠshowĠthatĠI^3Ġâī¤8Ġ*1Ġ*1âĨĴĠIĠâī¤2.ĊĊ
Rollout 14, Act: -1.654
ĠmeĠsee.ĊĊWeĠneedĠtoĠrelateĠI^3Ġâī¤8ĠJĠK.ĊĊByĠAM_ĠGM,ĊĊI
Rollout 14, Act: -1.702
WeĠneedĠtoĠshowĠthatĠ$I^3Ġ\leqĠ8ĠJĠK$.ĊĊHmm.ĠSo

DOWN_PROJ

Top 16 Positive Activations
Rollout 25, Act: 3.417
ĠwithĠ3ĠsignificantĠdigits.ĠUseĠ$6357Ġ\mathrm{~km}$ĠasĠtheĠradius
Rollout 62, Act: 2.553
663.ĠCheck.Ġ2663.ĠLikelyĠprime?ĊĊSimilarly,maybeĠitĠwillĠtake
Rollout 81, Act: 2.379
adraticĠsolution:Ġh=Ġ[-70Ġ±sqrt(70^2Ġ+4*11*
Rollout 81, Act: 2.335
QuadraticĠsolution:ĊĊh=(70Ġ±sqrt(4900Ġ+4*11*
Rollout 86, Act: 2.335
Ġpi/4,ĠcosxĠisĠgreaterĠthanĠsqrt(2)/2âīĪ0.707,
Rollout 7, Act: 2.329
16xĠ=71NĠ-209N/3=Ġ(213NĠ-
Rollout 25, Act: 2.326
ĠradiusĠofĠtheĠEarthĠisĠprovidedĠasĠ6357Ġkm.ĠTheĠquestionĠspecifiesĠthatĠweĠshouldĠconsider
Rollout 25, Act: 2.301
Ġsurface,ĠwhichĠisĠEarth'sĠradiusĠ6357ĠkmĠ+Ġ3541ĠkmĠ=
Rollout 38, Act: 2.284
ĠquadraticĠformula:ĊĊr=(625±sqrt(6252Ġ-4*168
Rollout 65, Act: 2.270
2/c=19952Ġ/sqrt(1995*a)=1995
Rollout 7, Act: 2.262
x.ĊĊTherefore,ĠtheĠequationĠbecomes:ĊĊ209N/3Ġ+16xĠ=71
Rollout 7, Act: 2.239
3=Ġ(213NĠ-209N)/3=4N/3.ĊĊTherefore
Rollout 28, Act: 2.221
2.ĠSoĠeigenvaluesĠareĠ(8Ġ±Ġsqrt(64Ġ-Ġ48))/2Ġ=Ġ(
Rollout 25, Act: 2.210
igeeĠequalĠtoĠEarth'sĠradiusĠ(6357Ġkm).ĠAssumeĠtheĠsameĠspeedĠatĠinfinityĠ(
Rollout 87, Act: 2.208
ĠinĠtheĠfirstĠquadrantĠupĠtoĠaĠradiusĠofĠsqrt(2)*a).ĠWait,ĠbutĠintegratingĠfromĠy
Rollout 95, Act: 2.204
ĠlengthĠP1Q1ĠwouldĠbeĠBC*sqrt(1/2450)Ġ=>Ġ(5
Top 16 Negative Activations
Rollout 26, Act: -2.557
ĠfromĠ$298.0Ġ\mathrm{~K}$ĠtoĠ$312.8Ġ\
Rollout 60, Act: -2.562
ĠsourceĠwithĠaĠspeedĠofĠ$2Ġ\mathrm{~ms}^{-1}$.ĠTheĠreflectedĠsignalĠis
Rollout 54, Act: -2.568
ĠethanethiolateĠ(HSâĪĴĠorĠCH2CH2SâĪĴ),ĠisĠaĠweakerĠbaseĠthan
Rollout 4, Act: -2.613
Ġ\pmĠ0.02Ġ\mathrm{~V}$.ĠWhatĠisĠtheĠpercentageĠerrorĠinĠthe
Rollout 82, Act: -2.667
ĠneedĠtoĠmodifyĠtheĠHuffmanĠalgorithmĠforĠaĠmixed-radixĠsystem?ĠLetĠmeĠrecallĠhowĠHuffmanĠcodingĠworks.
Rollout 48, Act: -2.671
ĠhexĠdumpĠwithĠspacesĠasĠgiven:Ċ4500Ġ0034ĠB612Ġ
Rollout 44, Act: -2.672
(\mathrm{p}_{1},Ġ\mathrm{~V}_{1},Ġ\mathrm{~T}_{
Rollout 60, Act: -2.694
f_{0}=492Ġ\mathrm{~Hz}$.ĠTheĠsoundĠisĠreflectedĠbyĠaĠlarge
Rollout 97, Act: -2.710
ĊGivenĠtheĠdecompositionĠreactionĠ\(2Ġ\mathrm{~N}_{2}Ġ\mathrm{O}_{5
Rollout 54, Act: -2.730
Ġ(OHâĪĴ)Ċ3.ĠPropionateĠ(CH2CH2COOâĪĴ)Ċ4.ĠMethanol
Rollout 97, Act: -2.746
ĠatĠanĠinitialĠpressureĠofĠ$1Ġ\mathrm{~atm}$.ĠAfterĠ$\mathrm{Y}
Rollout 48, Act: -2.753
ĠtheĠgivenĠhexĠdataĠis:Ġ4500Ġ0034ĠB612Ġ
Rollout 97, Act: -2.793
userĊTheĠdecompositionĠreactionĠ$2Ġ\mathrm{~N}_{2}Ġ\mathrm{O}_{5
Rollout 26, Act: -3.010
2}$ĠatĠ$298Ġ\mathrm{~K}$ĠandĠ1ĠatmĠintoĠ$\mathrm{
Rollout 27, Act: -3.016
Ġ10^{-17}Ġ\mathrm{~J}$,ĠcalculateĠtheĠrmsĠspeedĠofĠtheĠBe
Rollout 48, Act: -3.174
)ĠinĠtheĠIPĠheader:Ġ4500Ġ0034ĠB612Ġ

Layer 12

GATE_PROJ

Top 16 Positive Activations
Rollout 90, Act: 2.044
9*9=9,081,ĠsoĠcongruenceĠholds.ĠPerfect.ĊĊThus,Ġfor
Rollout 85, Act: 1.989
ĠSinceĠ(0,0)ĠisĠ0,ĠsoĠpossibleĠvaluesĠforĠtheĠotherĠtwoĠareĠ1Ġand
Rollout 90, Act: 1.951
1009ĠisĠ897,ĠsoĠNâī¡-M*897Ġmod1
Rollout 62, Act: 1.916
ĠYes,Ġ11*173.ĠSoĠtwoĠprimes.ĊĊm=45:20
Rollout 62, Act: 1.908
Ġ22ĠisĠ11*2.ĠSoĠtheĠresultĠisĠ473=11
Rollout 62, Act: 1.886
*181=1991.ĠSoĠtwoĠprimes.ĊĊHmm,Ġtedious.ĠSoĠmaybeĠthere
Rollout 3, Act: 1.883
ĠwhichĠisĠ300Ġ-Ġ2,ĠsoĠsubtractĠ66.512Ġ*Ġ
Rollout 90, Act: 1.881
ĠM=1,ĠN=112.ĠSoĠAĠisĠ112Ġ(N=1
Rollout 68, Act: 1.875
ĠatĠx=-2ĠisĠk=Ġ-delta,ĠsoĠdeltaĠpositiveĠapproachingĠzero.ĠSoĠf(-2)=
Rollout 48, Act: 1.868
Ġ0x50ĠisĠ80,ĠsoĠ2048Ġ+Ġ80Ġ+
Rollout 15, Act: 1.861
.ĊĊL1:ĠneighborĠL2Ġ(on).ĠSoĠexactlyĠoneĠneighborĠon.ĠTherefore,ĠturnsĠon.ĊĊ
Rollout 90, Act: 1.860
112*1=112.ĠSoĠN=112.ĠThat'sĠaĠthree
Rollout 56, Act: 1.852
ĠandĠA2A0ĠisĠâĪļ3,ĠsoĠA3A0Ġ=ĠâĪļ3Ġ/Ġ
Rollout 64, Act: 1.847
12=2009+3,ĠsoĠ2^{2012}Ġ-1
Rollout 62, Act: 1.839
7*79=1343.ĠSoĠtwoĠprimesĠagain.ĊĊm=38:Ġ
Rollout 49, Act: 1.834
.ĠIfĠeveryĠsubspaceĠisĠÏĥ-compact,ĠsoĠXĠisĠÏĥ-compact.ĠTherefore,ĠXĠis
Top 16 Negative Activations
Rollout 42, Act: -3.178
ĠdeterminingĠwhereĠtheĠdoubleĠbondĠforms.ĠSinceĠtheĠcentralĠcarbonĠisĠattachedĠtoĠtwoĠringsĠ(cyclobut
Rollout 54, Act: -3.196
ĠaccurateâĢĶprobablyĠbetterĠtoĠcompareĠinĠtermsĠofĠbaseĠstrength.ĠHydroxideĠisĠaĠmuchĠstrongerĠbaseĠthan
Rollout 42, Act: -3.222
ĠisĠaĠ-CH2OHĠgroupĠattachedĠtoĠaĠcarbonĠthatĠisĠconnectedĠtoĠbothĠaĠcyclobutyl
Rollout 54, Act: -3.224
Ġsolvated,ĠsoĠworstĠnucleophile.ĠRCOOâĪĴĠisĠweakestĠbase,ĠsoĠleastĠsolv
Rollout 54, Act: -3.233
1Ġreaction,ĠtheĠRDSĠisĠtheĠformationĠofĠcarbocation.ĠIfĠthatĠstepĠisĠRDS,Ġthen
Rollout 54, Act: -3.282
ĠweakerĠbaseĠthanĠhydroxide,ĠbutĠmaybeĠRCOOâĪĴĠisĠmoreĠhindered.ĠWait,Ġbut
Rollout 54, Act: -3.293
ĠsolventsĠ(fromĠbetterĠtoĠworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<ĠHOâĪĴ
Rollout 54, Act: -3.297
ĠthanĠtheĠreactant,ĠwhichĠlowersĠtheĠÎĶGâĢ¡ĠandĠspeedsĠupĠtheĠreaction.ĠIfĠincreasingĠpolarityĠstabil
Rollout 91, Act: -3.303
Wait,ĠagainĠsimilarĠsteps.ĠLet'sĠgoĠthroughĠcarefullyĠcalculatingĠangleĠC:ĊĊVectorĠCA=(tĠ-1
Rollout 54, Act: -3.310
ĠPropionateĠisĠaĠcarboxylate,ĠRCOOâĪĴ,ĠsoĠeachĠofĠthemĠexceptĠmethanol
Rollout 25, Act: -3.327
ĠrequireĠnumericalĠintegration,ĠbutĠperhapsĠapproximatingĠusingĠtheĠimpulseĠapproximation.ĠAssumeĠduringĠencounter,ĠtheĠtimeĠisĠshort
Rollout 42, Act: -3.345
utylĠandĠcyclopropyl),ĠtheĠadjacentĠcarbonsĠareĠpartĠofĠtheseĠrings.ĠWait,Ġno
Rollout 54, Act: -3.349
ĠforĠOĠnucleophilesĠinĠwaterĠisĠtypicallyĠRCOOâĪĴĠ<ĠHOâĪĴĠ<ĠROâĪĴ.ĠBecause
Rollout 42, Act: -3.358
ĠaĠcyclopropylĠgroup.ĠSoĠtheĠcentralĠcarbonĠhasĠthreeĠsubstituents:ĠoneĠhydroxyl
Rollout 54, Act: -3.371
ĠmightĠmakeĠitĠlessĠnucleophilicĠbecauseĠtheĠloneĠpairsĠareĠharderĠtoĠaccess,ĠbutĠinĠsolution,Ġmaybe
Rollout 54, Act: -3.658
icĠsolventsĠ(fromĠbetterĠtoĠworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<ĠHO

UP_PROJ

Top 16 Positive Activations
Rollout 71, Act: 2.084
ĠtheĠringĠofĠintegers,ĠdifferentĠfactorizationsĠmightĠgiveĠdifferentĠsolutions.ĠButĠgivenĠtheĠequationĠe2Ġ+3
Rollout 42, Act: 2.067
).ĠUponĠdepartureĠofĠwater,ĠaĠ**tertiaryĠcarbocation**ĠformsĠatĠtheĠcentralĠcarbon,Ġwhich
Rollout 42, Act: 2.065
ĠleavingĠgroupĠafterĠprotonationĠ(water),ĠperhapsĠE1ĠisĠpossible.ĠHowever,ĠsinceĠtheĠsubstrateĠisĠa
Rollout 42, Act: 2.039
ĠhydroxylĠgroup,ĠconvertingĠitĠintoĠaĠgoodĠleavingĠgroupĠ(water).ĠThen,ĠtheĠeliminationĠwouldĠoccur
Rollout 42, Act: 2.011
ĠcarbocationĠisĠatĠtheĠcentralĠcarbonĠ(tertiaryĠmaybe?),ĠperhapsĠadjacentĠtoĠaĠthree-memberedĠring
Rollout 49, Act: 2.001
,ĠbutĠisĠitĠhereditarilyĠLindelöf?ĠProbablyĠnot.ĠIfĠyouĠtakeĠaĠsubsetĠthat
Rollout 98, Act: 1.987
quenceĠstar.ĠLet'sĠthinkĠaboutĠwhatĠspectralĠtypeĠB5Ġimplies.ĠB5ĠstarsĠareĠmassive.ĠFor
Rollout 49, Act: 1.963
ĠisĠhereditarilyĠLindelö,ĠregularĠ(Hausdorff)ĠandĠitĠisĠaĠHer
Rollout 49, Act: 1.961
ĠuncountableĠandĠhereditariyĠLindelof,ĠthenĠitĠhasĠaĠuncountablesubsetĠwith
Rollout 14, Act: 1.961
ĠofĠintegralsĠandĠuseĠsomeĠsortĠofĠdualĠHölderĠapplications?ĊĊWait,Ġlet'sĠsee.ĠLet's
Rollout 42, Act: 1.960
ocation.ĠWait,ĠbutĠifĠtheĠmechanismĠisĠE1,ĠthenĠweĠfirstĠgetĠcarbocationĠformation.ĠHowever
Rollout 42, Act: 1.929
ĠaĠgoodĠacid,Ġit'sĠmoreĠlikelyĠanĠE1Ġmechanism.ĠButĠsometimes,ĠwithĠpoorĠleavingĠgroups,
Rollout 42, Act: 1.914
Ġgroups:ĠtwoĠringsĠandĠaĠhydrogen?),ĠsoĠmaybeĠtertiaryĠalcoholsĠareĠmoreĠproneĠtoĠE1.
Rollout 49, Act: 1.909
,ĠwhichĠwouldĠcontradictĠtheĠhereditaryĠLindelöfĠpropertyĠ(sinceĠaĠdiscreteĠuncountableĠsubspace
Rollout 42, Act: 1.903
xylĠgroupĠisĠprotonated,ĠformingĠaĠgoodĠleavingĠgroupĠ(water).ĠUponĠdepartureĠofĠwater,Ġa
Rollout 42, Act: 1.887
ĠisĠpossible.ĠHowever,ĠsinceĠtheĠsubstrateĠisĠaĠtertiaryĠalcoholĠ(theĠcentralĠcarbonĠisĠattachedĠtoĠthreeĠcarbon
Top 16 Negative Activations
Rollout 91, Act: -1.634
|Ġ)Ġ=Ġ2tĠ/Ġ(2Ġsqrt(Ġ(tĠ-1)^2Ġ+Ġt2Ġ))
Rollout 65, Act: -1.642
ĠInsight:ĊĊWhenĠyouĠsplitĠa.rectangleĠintoĠtwoĠSimilar.ones,ĠtheĠraitoĠisĠsquares.ĊĊĠSo,
Rollout 56, Act: -1.650
Ġincrement:ĊĊÎĶx=Ġ(-14)/sqrt(559)Ġ*Ġsqrt(559
Rollout 91, Act: -1.655
1Ġ}Ġ*Ġ2Ġ)Ġ=ĠtĠ/Ġsqrt{2t2Ġ-2tĠ+1Ġ}ĊĊSo
Rollout 95, Act: -1.658
50Ġ=ĠBC,ĠwhichĠhasĠlength5Ġsqrt(3),Ġirrational,æīĢä»¥ä¹Łä¸įrationalãĢĤĊĊThus
Rollout 95, Act: -1.669
ĠtheĠproblemâĢĻsĠoriginalĠBCĠlengthĠisĠ5Ġsqrt(3),ĠperhapsĠthereĠisĠanĠerrorĠinĠpriorĠcoordination
Rollout 95, Act: -1.671
ĠtotalĠlengthĠisĠPkQk=ĠkĠsqrt(3)/(2*245Ġ))Ġ=Ġ(
Rollout 56, Act: -1.712
ĠMovingĠalongĠtheĠdirectionĠ(Ġ(-14)/sqrt(559),Ġ(11âĪļ3)/
Rollout 53, Act: -1.715
x_CĠ)Ġ/2.Ġx_BĠisĠsqrt(73Ġ)Ġ/2.ĠSo:ĊĊx_B
Rollout 53, Act: -1.719
.5:ĠdistanceĠfromĠpointĠP(c/3,Ġd/3)ĠtoĠE(0,0
Rollout 95, Act: -1.727
450Ġ)ĊĊĠDistanceĠbetweenĠisĠ(9Ġsqrt(671))/36750.ĊĊ
Rollout 81, Act: -1.732
lying:ĊĊ(-51Ġ+Ġh2Ġ)sqrt(533Ġ)=Ġ7*sqrt(
Rollout 53, Act: -1.753
ĠmidpointĠofĠBC.ĊĊCoordinatesĠofĠB:Ġ(Ġsqrt(73)/2,Ġ0Ġ)ĊĊCoordinatesĠof
Rollout 53, Act: -1.762
C.ĊĊFirst,Ġpoints:ĊĊA:Ġ(-sqrt(73)/2,Ġ0)ĊĊE:
Rollout 53, Act: -1.774
Ġ2:ĠdistanceĠfromĠpointĠP(c/3,Ġd/3)ĠtoĠD(Ġ(aĠ+
Rollout 53, Act: -1.841
3Ġ)Ġ/2.ĠSo:ĊĊx_BĠsqrt(73Ġ)Ġ/2ĠâīĪĠsomething.ĠAdd

DOWN_PROJ

Top 16 Positive Activations
Rollout 80, Act: 2.729
Ġ10^{-14}Ġ\mathrm{~J}$ĠandĠreachesĠaĠfluxĠdensityĠofĠ$1
Rollout 25, Act: 2.687
ĠdistanceĠofĠ$3541Ġ\mathrm{~km}$ĠtoĠEarth'sĠsurfaceĠwithĠaĠspeedĠof
Rollout 60, Act: 2.679
f_{0}=492Ġ\mathrm{~Hz}$.ĠTheĠsoundĠisĠreflectedĠbyĠaĠlarge
Rollout 27, Act: 2.627
Ġ10^{-17}Ġ\mathrm{~J}$,ĠcalculateĠtheĠrmsĠspeedĠofĠtheĠBe
Rollout 17, Act: 2.611
Ġ10^{-26}Ġ\mathrm{~J}Ġ/Ġ\mathrm{T}$.ĠLetĠthe
Rollout 4, Act: 2.607
Ġ\pmĠ0.02Ġ\mathrm{~V}$.ĠWhatĠisĠtheĠpercentageĠerrorĠinĠthe
Rollout 97, Act: 2.559
userĊTheĠdecompositionĠreactionĠ$2Ġ\mathrm{~N}_{2}Ġ\mathrm{O}_{5
Rollout 26, Act: 2.405
2}$ĠatĠ$298Ġ\mathrm{~K}$ĠandĠ1ĠatmĠintoĠ$\mathrm{
Rollout 97, Act: 2.310
ĊGivenĠtheĠdecompositionĠreactionĠ\(2Ġ\mathrm{~N}_{2}Ġ\mathrm{O}_{5
Rollout 80, Act: 2.218
ĠofĠradiusĠ$1200Ġ\mathrm{~km}$,ĠdensityĠ$12.8Ġ\
Rollout 44, Act: 2.214
(\mathrm{p}_{1},Ġ\mathrm{~V}_{1},Ġ\mathrm{~T}_{
Rollout 17, Act: 2.195
Ġ10^{-11}Ġ\mathrm{~m}$ĠisĠtheĠBohrĠradius.ĠEvaluateĠthe
Rollout 56, Act: 2.190
0A_2Ġ=Ġ\sqrt{3}Ġ\),ĠandĠ\(ĠA_1A_2Ġ=
Rollout 97, Act: 2.105
ĠatĠanĠinitialĠpressureĠofĠ$1Ġ\mathrm{~atm}$.ĠAfterĠ$\mathrm{Y}
Rollout 63, Act: 2.078
ĠsidesĠbyĠ\(Ġ\frac{4}{3}Ġ\):ĊĊ\[Ċf(2,Ġ1)
Rollout 84, Act: 2.073
ĠscenarioĠisĠ\(Ġ\frac{2}{9}Ġ\)ĠofĠtheĠoriginalĠwastedĠamount.ĠSettingĠupĠthe
Top 16 Negative Activations
Rollout 14, Act: -1.825
Ġ+ĠK/A.ĊĊPlugĠA=2K/Ġsqrt(J):ĊĊFirstĠtermĠsqrt(J)*Ġsqrt(A)=
Rollout 64, Act: -1.839
Ġz>=0,ĠtheĠnumberĠofĠintegerĠsolutionsĠisĠfloor(C)Ġ+1,ĠsinceĠz=0,
Rollout 49, Act: -1.850
ĠEveryĠregularĠLindelöfĠspaceĠisĠparacompactĠandĠfullyĠnormal.ĠButĠweĠneedĠtheĠconverse
Rollout 57, Act: -1.851
ĠtheĠusualĠtopology,ĠandĠfailsĠtoĠbeĠparacompactĠorĠsuch,ĠwhichĠrelatesĠtoĠnormalityĠin
Rollout 97, Act: -1.856
Ġis\boxed{2.30}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{230}ĊĊ
Rollout 65, Act: -1.882
)/2=41.ĠButĠsubtractingĊĊ**FinalĠAnswer**Ċ\boxed{40}ĊĊ<|im_start|>
Rollout 99, Act: -1.888
573âīĪ(-0.8235)/1.1573âīĪ-0
Rollout 57, Act: -1.904
ĠregularĠbutĠnotĠnormal.ĠčĊčĊ*A.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġso
Rollout 65, Act: -1.909
ĠorĠequalĠto1995ĠisĠequalĠtoĠceilingĠhalfĠtheĠnumberĠofĠtotalĠdivisorsĠsinceĠtheyĠcome
Rollout 62, Act: -1.920
ĠanswerĠisĠ\boxed{132}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{132}ĊĊ
Rollout 57, Act: -1.926
ĠsubsetsĠofĠtheĠrealĠlineĠthatĠareĠbothĠF-sigmaĠsetsĠinĠtheĠEuclideanĠtopologyĠbutĠareĠnotĠsepar
Rollout 17, Act: -1.957
e-11)^3Ġ)ĊĊTheĠ4ÏĢe-7ĠisĠμ0,ĠwhichĠisĠ
Rollout 49, Act: -1.972
ĠaĠLindelöfĠspace,ĠhenceĠparacompact.ĠAndĠthenĠinĠaĠlocallyĠcompactĠparac
Rollout 99, Act: -2.088
75ĠâīĪĠ(-0.8235)/1.1575ĠâīĪĠ-
Rollout 49, Act: -2.101
ĠspaceĠisĠLindelöfĠandĠthusĠparacompact,ĠbutĠnotĠnecessarilyĠsecond-countable.ĊĊAlternatively
Rollout 49, Act: -2.243
act.ĠAndĠthenĠinĠaĠlocallyĠcompactĠparacompactĠHausdorffĠspace,Ġit'sĠthe

Layer 13

GATE_PROJ

Top 16 Positive Activations
Rollout 37, Act: 3.317
Ġproducts,ĠbutĠmaybeĠinapplicable.ĊĊWait,ĠthinkĠonĠtheĠoriginalĠproblem,ĠwhetherĠtheĠdeterminantĠstructureĠcan
Rollout 37, Act: 3.292
ĠgroupedĠunderĠcongruenceĠmoduloĠn!.ĊĊWait,ĠthinkĠofĠitĠthisĠway:ĠTheĠsetĠofĠtermsĠin
Rollout 12, Act: 3.255
Ġ[P.ĠTuran]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠproveĠthatĠthe
Rollout 63, Act: 3.250
ĠforĠtheĠthree-variableĠcase.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠaĠfunction
Rollout 1, Act: 3.245
mathcal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠthere
Rollout 6, Act: 3.217
ĠhashĠofĠα_n.ĠNotĠsure.ĊĊWait,ĠthinkĠaboutĠP_angel:ĠtheĠoriginalĠTMĠMĠis
Rollout 51, Act: 3.199
)|\Ġdx.Ġ\]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠintegralĠtoĠsolve
Rollout 45, Act: 3.196
ĠofĠĠ$Ġm$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠproblemĠwhereĠa
Rollout 11, Act: 3.166
1/cĠ=ĠbcĠ+ĠacĠ+Ġab.ĠSoĠpĠcanĠbeĠwrittenĠasĠ(aĠ+ĠbĠ+Ġc
Rollout 14, Act: 3.118
ĠdxĠ\right)Ġ$$ĠĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠtoĠproveĠthisĠinequality
Rollout 28, Act: 3.110
ĠofĠtheseĠhyperbolas.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthisĠproblemĠhere
Rollout 83, Act: 3.106
ĠeachĠassignedĠtoĠaĠvertex.ĠTheĠsumĠconditionĠappliesĠcyclically,ĠsinceĠit'sĠaĠnonagon,Ġso
Rollout 66, Act: 3.089
Ġ.ĠFindĠ$p$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthis
Rollout 17, Act: 3.076
$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackleĠthisĠproblem
Rollout 69, Act: 3.075
ĠTeamĠRoundĠ#10*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠpositiveĠintegers
Rollout 11, Act: 3.070
)Ġ+Ġ(abĠ+ĠbcĠ+Ġca).ĠSoĠpĠ=ĠaĠ+ĠbĠ+ĠcĠ+ĠabĠ+Ġbc
Top 16 Negative Activations
Rollout 64, Act: -1.455
ĠasĠhighĠasĠ2^{2012Ġ).ĊĊButĠinĠreality,ĠforĠy=0,Ġwhen
Rollout 81, Act: -1.461
11,ĠyieldingĠapprox72.8Ġdegrees?ĊĊThisĠisĠaĠconflict.ĊĊWait,ĠsoĠseemsĠthere
Rollout 65, Act: -1.469
ĠifĠa=ĠRĠ=1995âĪļ.ĊĊButĠa=ĠsqrĠ1995:
Rollout 65, Act: -1.470
ĠTherefore,ĠinitiallyĠthisĠapproachĠonlyĠsuggestsĠ1Ġtriple.ĊĊButĠtheĠproblemĠsaysĠ"orderedĠtriplesĠ(a,b
Rollout 20, Act: -1.470
ĠLCMĠofĠ43ĠandĠ47)?ĊĊWait,ĠbutĠneedĠtoĠcheckĠcarefully.ĠWait,
Rollout 97, Act: -1.476
3Ġs,ĠsoY=2.30.ĊĊButĠproblemĠasĠitĠisĠposed.ĠHeĠmightĠassume
Rollout 52, Act: -1.476
,ĠwhichĠisĠwayĠlowerĠthanĠpriorĠ-11.ĊĊProceeding.ĊĊ10.Ġk=10
Rollout 33, Act: -1.482
,ĠsoĠthirdĠpositionĠis4.ĠFourthĠpositionĠremaining.ĊĊSoĠpermutationsĠstartingĠwith2,3,4.ĊĊ
Rollout 98, Act: -1.487
Ġ(let'sĠsupposeĠ7ĠsolarĠmasses,Ġmaybe).ĠButĠsuchĠaĠstarĠwouldĠendĠasĠaĠsupernova
Rollout 0, Act: -1.493
,ĠtheĠformulaĠisĠ2^{kĠ-1).ĊĊHence,ĠforĠ20!ĠwithĠ
Rollout 99, Act: -1.495
Ġx3ĠisĠ2.157Ġapproximately.ĊĊHowever,ĠsinceĠinĠtheĠproblemĠtheyĠsayĠtoĠperform
Rollout 99, Act: -1.498
ĠwhileĠmentioningĠtwoĠiterations,ĠmaybeĠtheyĠexpectĠanotherĠnotation.ĊĊTherefore,ĠGivenĠthisĠconfusion,ĠbutĠgivenĠpreciseĠcalculation
Rollout 3, Act: -1.513
69.ĠMaybeĠ168Ġm/s.ĊĊButĠcheckingĠfurtherĠonĠspeed,ĠconsideringĠtheĠproblemĠmight
Rollout 26, Act: -1.534
ĠelseĠperhapsĠoneĠdecimalĠasĠ90.4.ĊĊWait.Ġ61.32ĠisĠprecise
Rollout 89, Act: -1.535
ĠthatĠchildrenĠcanĠbeĠparts?ĠNo,ĠobviouslyĠnot.ĊĊAlternativeĠIdea:ĠIfĠthisĠisĠaĠcertainĠclass,
Rollout 7, Act: -1.602
ĠconstraintsĠmissed.ĠAlternatively,ĠperhapsĠN=24?ĊĊLetĠmeĠcheckĠmanuallyĠforĠN=24.ĊĊ

UP_PROJ

Top 16 Positive Activations
Rollout 35, Act: 2.885
ĠonlyĠtwoĠintegerĠsolutions,ĠR(n)ĠmustĠbeĠdesignedĠsoĠthatĠexceptĠforĠn=19ĠandĠ
Rollout 87, Act: 2.679
Ġa,ĠzĠ=ĠtĠa.ĠThus,ĠeachĠfromĠ0ĠtoĠ1.ĊĊButĠreturningĠtoĠsubstitution
Rollout 65, Act: 2.655
/nĠtoĠbeĠinteger,ĠbutĠnĠmustĠbeĠatĠleastĠ2ĠforĠk<1.ĠTherefore,Ġthis
Rollout 65, Act: 2.605
1995).ĠButĠcĠmustĠbeĠatĠleast1995,ĠsoĠifĠa=1
Rollout 20, Act: 2.542
ĠanyĠpositiveĠinteger.ĠSo,ĠifĠkĠisĠallowedĠtoĠbeĠanyĠinteger,ĠthenĠtheĠrequirementĠisĠthatĠfor
Rollout 72, Act: 2.540
ĠofĠthreeĠrootsĠmustĠbeĠreal,ĠhenceĠit'sĠnecessaryĠthatĠeitherĠtheyĠareĠallĠreal,ĠorĠtheĠimaginary
Rollout 90, Act: 2.532
1009.ĊĊButĠsinceĠMĠisĠaĠsingleĠdigit,ĠthenĠ(-9N)Ġmod10
Rollout 2, Act: 2.527
1000.ĠAs997ĠtermsĠfromk=3Ġtok=999.Ġ
Rollout 19, Act: 2.518
.ĠHowever,Ġ\(ĠkĠ\)ĠmustĠbeĠatĠleastĠ6Ġ(sinceĠ\(ĠnĠ\)ĠisĠa
Rollout 65, Act: 2.513
.e.,ĠexponentĠatĠleastĠ1,ĠwhichĠgivenĠthatĠa2ĠhasĠexponentsĠeven,ĠexponentsĠare
Rollout 0, Act: 2.462
?ĠNo,Ġthat'sĠnotĠcorrect.ĠEachĠmustĠhaveĠallĠprimesĠexponentsĠfromĠ20!ĠFor
Rollout 90, Act: 2.437
2018,ĠsoĠAĠmustĠbeĠatĠleastĠ2018.ĠHowever,ĠAĠmust
Rollout 15, Act: 2.413
'tĠboundĠbyĠmod2.ĠButĠwhenĠyouĠtrackĠbasedĠonĠrulesĠ(lampĠisĠONĠtomorrowĠonlyĠifĠexactly
Rollout 90, Act: 2.384
ĠM,ĠbutĠtheĠproblemĠspecifiesĠthatĠMĠisĠaĠsingleĠdigit,Ġ0-9.ĠTherefore,Ġonly
Rollout 20, Act: 2.379
forĠallĠpositiveĠintegersĠa".ĠWhenĠaĠisĠallowedĠtoĠbeĠanyĠpositiveĠinteger,ĠweĠhaveĠtoĠconsider,
Rollout 65, Act: 2.370
Similarly,ĠtryingĠotherĠpermutations,ĠbutĠperhapsĠwithĠeachĠother.ĊĊButĠthinkingĠagain,ĠinĠtermsĠofĠcuttingĠalong
Top 16 Negative Activations
Rollout 12, Act: -1.923
ĠisĠmonotonicallyĠdecreasingĠinĠtheĠintervalĠĠ$Ġ0<\nu<1$Ġ.Ġ[P
Rollout 55, Act: -1.929
{A}^{2018}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI
Rollout 86, Act: -1.930
2}}f(x)\Ġdx.$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠtheĠproblemĠisĠasking
Rollout 18, Act: -1.943
ĠsuchĠthatĠĠ$H=H_{xy}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet's
Rollout 69, Act: -2.005
ĠinĠdegrees,ĠcomputeĠĠ$a+b$Ġ.ĊĊĊ*2021ĠCCAĠMathĠBonanza
Rollout 77, Act: -2.015
ĠCalculateĠ$f(14,52)$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 90, Act: -2.029
ĠareĠdivisibleĠbyĠĠ$2018$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI
Rollout 71, Act: -2.043
a|Ġ+Ġ|b|Ġ+Ġ|c|$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 12, Act: -2.043
ĠĠ$Ġ0<\nu<1$Ġ.Ġ[P.ĠTuran]Ċ<|im_start|>assistantĊ<|im_start|>
Rollout 72, Act: -2.092
ĠĠwhoseĠsumĠisĠĠ$ĠĊ1.Ġ$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet's
Rollout 71, Act: -2.144
Ġ,Ġ$b$Ġ,ĠandĠ$c$Ġ.ĠĠFindĠ$|a|Ġ+Ġ|b|Ġ+
Rollout 72, Act: -2.145
.Ċ<|im_start|>userĊFindĠallĠnumbersĠĠ$ĠnĠ$ĠĠforĠwhichĠthereĠexistĠthreeĠ(notĠnecessarilyĠdistinct
Rollout 72, Act: -2.147
ĠĠ$ĠnĠ$ĠĠwhoseĠsumĠisĠĠ$ĠĊ1.Ġ$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊ
Rollout 51, Act: -2.147
sinĠ(x-\theta)|\Ġdx.Ġ\]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 1, Act: -2.170
ĠvectorsĠinĠĠ$Ġ\mathcal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 86, Act: -2.261
{\pi}{2}}f(x)\Ġdx.$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠthe

DOWN_PROJ

Top 16 Positive Activations
Rollout 30, Act: 2.087
1Ġ).ĊĊTherefore,Ġ(b_nĠ+1Ġ)ĠtermĠcanĠbeĠwrittenĠasĠaĠratioĠofĠa_{n
Rollout 3, Act: 1.985
ĠLetĠmeĠmakeĠsure.ĊĊAccordingĠtoĠMaxwell-BoltzmannĠdistribution,ĠtheĠmeanĠspeedĠisĠtheĠaverageĠof
Rollout 73, Act: 1.954
Ġ-Ġ2046=16388.ĊĊTherefore,ĠgoingĠbackĠtoĠN:ĊĊN
Rollout 87, Act: 1.928
ĠnumericalĠconstants,ĠsoĠanswerĠwouldĠbeĠexpressionĠlikeĠ\(Ġ\frac{\rhoĠa^2}{\epsilon_
Rollout 64, Act: 1.893
ĠfromĠnumerator:ĊĊ=2^{2009}[8Ġ+a1*(1Ġ-2/2
Rollout 3, Act: 1.810
Ġ(average)ĠspeedĠderivedĠfromĠtheĠMaxwell-BoltzmannĠdistribution:ĊĊ\[ĊvĠ=Ġ\sqrt{\
Rollout 26, Act: 1.684
s)$:ĊĠĠĠ\[ĊĠĠĠ\DeltaĠH_f^\circĠ(\mathrm{HgO})Ġ=
Rollout 42, Act: 1.646
ocationĠformation.ĠHowever,ĠifĠtheĠcarbocationĠisĠadjacentĠtoĠstrainedĠrings,ĠlikeĠcyclopropane,
Rollout 91, Act: 1.641
Ġ=Ġ(tĠ+1)/sqrt(2Ġt2Ġ+Ġ2ĠtĠ+1Ġ)ĊĊSimilarly,Ġsin
Rollout 10, Act: 1.634
ĠtheoryĠhere.ĊĊRecallĠbyĠLegendre'sĠthree-squareĠtheorem,ĠaĠnumberĠnĠisĠexpressibleĠasĠa
Rollout 3, Act: 1.624
8ĠKĠ(unlessĠstatedĠotherwise),ĠperhapsĠanswerĠaroundĠ\boxed{169}Ġm/s.ĠBut
Rollout 62, Act: 1.621
121:ĠcomputeĠexpression=1212-121Ġ+11=14
Rollout 75, Act: 1.618
ĠdependsĠonĠp.ĊĊButĠbyĠLegendre'sĠthree-squareĠtheorem,ĠaĠpositiveĠintegerĠnĠcanĠbeĠexpressedĠas
Rollout 99, Act: 1.608
0.8027467Ġz2Ġ-0.7993134
Rollout 81, Act: 1.590
ĠangleĠbetweenĠABĠandĠAC,ĠsoĠtheĠactualĠangle'sĠtangentĠisĠpositive,ĠsoĠperhapsĠtheĠabsoluteĠvalueĠapplies
Rollout 14, Act: 1.590
HS=1ĠandĠRHS=4.ĠSomething'sĠnotĠaddingĠupĠhere.ĠWait,ĠbutĠtheĠquestionĠstates
Top 16 Negative Activations
Rollout 95, Act: -2.871
)/15Ġ+Ġ0Ġ=Ġ(9Ġsqrt(671))/36750.ĊĊ
Rollout 95, Act: -2.871
Ġ(Ġ-2/3)^2Ġ+Ġ(Ġsqrt(671)/3Ġ)^2Ġ)=
Rollout 53, Act: -2.892
)/2))=Ġ-27/(2Ġsqrt(73))=ĠmultipliedĠbyĠsqrt(73
Rollout 95, Act: -2.914
)ĠtoĠC(25/3,Ġsqrt(671)/3).ĊĊĠvectorĠcomponents:2
Rollout 53, Act: -2.918
73Ġ)Ġ=Ġ-27/(2Ġsqrt(73))Ġ)ĠRationallze:Ġthis
Rollout 72, Act: -2.920
^2)=1Ġ+Ġ2*(-(sqrt(5)+1)/4)=1Ġ-Ġ(sqrt
Rollout 53, Act: -2.928
Ġas:ĊĊc=Ġ-27/(2Ġsqrt(73)),ĠandĠinĠequation3:ĊĊc2
Rollout 95, Act: -2.934
450Ġ)ĊĊĠDistanceĠbetweenĠisĠ(9Ġsqrt(671))/36750.ĊĊ
Rollout 95, Act: -2.955
)ĠtoĠC(25/3,Ġsqrt(671)/3),ĊĊĠSoĠAQkĠis
Rollout 95, Act: -2.958
3675Ġ)=Ġ(15Ġsqrt(3))/(2*3675Ġ)=
Rollout 95, Act: -2.959
7/3=Ġ(-2/3,Ġsqrt(671)/3Ġ)Ġ,lengthĠsqrt(
Rollout 95, Act: -2.964
ĠupĠtoĠ(25/3Ġa,Ġsqrt(671)/3ĠaĠ),ĠadndĠlength
Rollout 95, Act: -3.017
ĠtheĠproblemâĢĻsĠoriginalĠBCĠlengthĠisĠ5Ġsqrt(3),ĠperhapsĠthereĠisĠanĠerrorĠinĠpriorĠcoordination
Rollout 53, Act: -3.025
,ĠthenĠc=-27/(4*(sqrt(73)/2))=Ġ-27/(
Rollout 95, Act: -3.043
)/3675=Ġ(15Ġsqrt(3))/(2*3675)=Ġsqrt
Rollout 95, Act: -3.081
è¿Ļæł·çļĦè¯Ŀ,k=490*(c/sqrt(3))*sqrt(3Ġ)=490

Layer 14

GATE_PROJ

Top 16 Positive Activations
Rollout 60, Act: 1.527
98ĊĊTherefore,Ġ492Ġ*Ġ(332/328)Ġ=Ġ
Rollout 25, Act: 1.385
_finalĠ=Ġ0ĊĊSo,ĊĊ2Ġμ_finalĠr_p_finalĠ=Ġv2Ġ(r2Ġ-Ġr
Rollout 29, Act: 1.339
.ĠSinceĠit'sĠopen,ĠgaugeĠpressureĠisĠrho*g*h.ĠMeanwhile,ĠinĠtheĠhandle,ĠtheĠpressure
Rollout 29, Act: 1.334
ĠpressureĠatĠtheĠbottomĠofĠtheĠhandleĠwouldĠbeĠrho*g*H.ĠThenĠtheĠgaugeĠpressureĠwouldĠbeĠrho
Rollout 29, Act: 1.316
*H.ĠThenĠtheĠgaugeĠpressureĠwouldĠbeĠrho*g*HĠ-ĠatmosphericĠpressure.ĠButĠsinceĠtheĠmain
Rollout 60, Act: 1.287
,Ġf'Ġ=Ġ492ĠHzĠ*Ġ(330Ġ+Ġ2)/330
Rollout 78, Act: 1.280
_wĠ+Ġ60ĊĊSo:ĊĊLĠ/Ġ(v_wĠ-Ġ20)Ġ=Ġ1
Rollout 78, Act: 1.272
Ġ40)ĠinĠtheĠequation:ĊĊLĠ/Ġ(v_wĠ-Ġ20)Ġ=Ġ1
Rollout 60, Act: 1.269
vĠ-Ġv_car)Ġ)Ġ=Ġf0Ġ*Ġ(vĠ+Ġv_car)/(vĠ-Ġv_car)ĊĊ
Rollout 29, Act: 1.261
ĠpressureĠatĠtheĠbottomĠofĠtheĠhandleĠwouldĠbeĠrho*g*H,ĠbutĠifĠtheĠmainĠjugĠisĠconnected
Rollout 29, Act: 1.255
ĠmainĠjug'sĠbottomĠhasĠaĠgaugeĠpressureĠofĠrho*g*h,ĠwhileĠtheĠhandle'sĠbottomĠ(ifĠconsidered
Rollout 25, Act: 1.254
ĊĊThus,ĊĊÎĶdâīĪĠ(1/2)a_meanĠt2Ġ=Ġ(1/2)*(ĠG
Rollout 29, Act: 1.253
ĠpressureĠofĠtheĠsealedĠairĠinĠtheĠhandleĠplusĠrho*g*H.ĠSinceĠtheĠpressureĠatĠtheĠinterfaceĠbetween
Rollout 29, Act: 1.235
ĠtheĠbottomĠofĠtheĠmainĠjugĠwouldĠstillĠbeĠrho*g*hĠ(gaugeĠpressure,ĠsinceĠit'sĠopen
Rollout 13, Act: 1.228
1936ĊĊTherefore,ĠA_annualĠ=Ġ6000Ġ*Ġ1.5
Rollout 29, Act: 1.224
ĠsituationĠwhereĠtheĠgaugeĠpressureĠatĠtheĠbottomĠisĠrho*g*hĠregardlessĠofĠtheĠhigherĠhandleĠlevel?ĠIfĠthe
Top 16 Negative Activations
Rollout 52, Act: -3.836
028ĠisĠtheĠmaximumĠk.ĠSoĠasĠsuch,ĠifĠweĠthinkĠthisĠthrough,ĠnĠisĠ(
Rollout 68, Act: -3.849
ĠbeĠnegativeĠ(sinceĠkĠnegative,ĠxĠnegative:ĠthusĠkxĠpositive).ĠButĠourĠdomainĠwasĠindeedĠx
Rollout 89, Act: -3.851
ĠbeĠsplit.ĠWhichĠwouldĠhaveĠbeenĠanĠissue.ĠThus,ĠtheĠanswer.ĠMaybeĠtheĠproblemĠdoesĠnotĠspecify
Rollout 15, Act: -3.852
Ġn=1=2^0.ĠLet'sĠthinkĠofĠourĠn=Ġ2^k.ĊĊFor
Rollout 89, Act: -3.854
.ĊĊButĠtheĠconventionalĠquestionĠwouldĠlikelyĠstateĠintegers.ĠHence,ĠwithĠthatĠgiven,ĠĠifĠyouĠtakeĠT
Rollout 20, Act: -3.855
ĠbutĠneedĠtoĠcheckĠcarefully.ĠWait,Ġlet'sĠthink:ĊĊButĠforĠmodulusĠ43:ĠnĠâī
Rollout 75, Act: -3.860
16Ġ+16Ġ+0Ġagain.ĊĊThisĠleadsĠmeĠtoĠthinkĠthatĠthereĠisĠanĠedgeĠcaseĠwhen
Rollout 62, Act: -3.860
ĠvaluesĠofĠmĠwhereĠtheĠquadraticĠexpressionĠisĠlarger,ĠhenceĠmoreĠlikelyĠtoĠbeĠcompositeĠwithĠmultipleĠfactors.ĊĊLet
Rollout 72, Act: -3.868
Ġ"numberĠn"ĠcouldĠreferĠtoĠpositiveĠintegers.ĠThus,ĠshouldĠweĠanswerĠasĠinĠtheĠproblem'sĠcontext
Rollout 85, Act: -3.940
.ĠForĠexample,Ġlet'sĠsupposeĠweĠstartĠalternatelyĠlabeling.ĊĊFromĠ(0,0)=0,
Rollout 89, Act: -3.947
Ġit'sĠimpossibleĠforĠtheĠprimaryĠschoolĠmathĠproblem.ĠThus,ĠperhapsĠmadeĠaĠmistake.ĠButĠinĠlieuĠof
Rollout 41, Act: -3.950
ĠtreatingĠasĠfour-digitĠwithĠleadingĠzeros:ĠcanĠalsoĠthinkĠofĠdifferentĠway.ĊĊTotalĠcountedĠwhenĠallowingĠleadingĠzeros
Rollout 37, Act: -3.953
Ġ-a1)ĠandĠforĠeachĠvariable,ĠbutĠhmm.ĊĊWait,ĠperhapsĠthereĠisĠmoreĠstructureĠhere.
Rollout 93, Act: -4.068
ĠconsideredĠasĠsomeĠcriticalĠpoints.ĠHmm.ĠI'mĠthinkingĠofĠusingĠOrthogonalĠPolynomialsĠmachineryĠhereĠbutĠI
Rollout 97, Act: -4.084
ĠviaĠODEĠandĠstoichĠintegration.ĠHmmâĢĶthusĠintegratingĠviaĠthisĠapproachĠtoĠtotalĠpressureĠgetsĠsameĠresult
Rollout 68, Act: -4.119
)Ġ]Ġ/Ġ2.ĊĊButĠlet'sĠthinkĠalternatively.ĠForĠwhenĠquadraticĠhasĠtwoĠroots,ĠmaybeĠwe

UP_PROJ

Top 16 Positive Activations
Rollout 99, Act: 2.239
ĠwasĠapproximately1.1573,ĠaddingĠtoĠx2=1,ĠweĠgetx3âīĪ
Rollout 64, Act: 2.171
}Ġ+1Ġ)ĊĊButĠlikely.ĠOrĠinĠotherĠterms,ĠcouldĠwriteĠthisĠas9*4*2
Rollout 74, Act: 2.140
.ĠSoĠyes,ĠthatĠseemsĠright.ĠSoĠcombiningĠwithĠtheĠfore-and-aftĠcomponents,ĠwhichĠwhenĠresolved
Rollout 97, Act: 2.135
1.45Ġatm.ĠButĠifĠtheĠaboveĠisĠtheĠcaseĠ(whichĠaĠfirst-orderĠdecayĠofĠN
Rollout 89, Act: 2.077
ĠtheĠexactĠdecimalĠ0.4,ĠbutĠthatĠalsoĠwouldĠrequireĠTĠsuchĠthatĠ0.4*T
Rollout 65, Act: 2.062
Ġk*c,ĠcĠ=Ġk*a,ĠbutĠthatĠalsoĠseemsĠtricky.ĊĊAlternatively,ĠmaybeĠit'sĠpossibleĠthat
Rollout 92, Act: 2.054
ĠsideĠasĠtheĠobject).ĠĊĊSo,ĠpluggingĠintoĠtheĠlensĠformula:Ġ1/fĠ=Ġ1
Rollout 1, Act: 2.040
ĠproblemĠisĠoverĠtheĠHilbertĠspace,ĠsoĠthinkingĠasĠinĠR^n,ĠforĠwhichĠweĠknowĠthatĠeven
Rollout 57, Act: 2.040
ĠC,ĠthenĠsinceĠtheĠcomplementĠofĠCĠisĠopenĠandĠcontainsĠx,ĠthereĠexistsĠaĠmeasurableĠsetĠAĠ(
Rollout 81, Act: 2.030
(angleĠBAD)*tan(angleĠCAD)].ĊĊThat'sĠfromĠtheĠtangentĠadditionĠformula.ĊĊGivenĠangleĠBAC=
Rollout 39, Act: 2.019
89Ġ(whichĠweĠcalculatedĠearlier).ĠSoĠinsertingĠinĠaĠnumberĠintoĠtheĠend,Ġyes,ĠifĠyou
Rollout 67, Act: 2.006
.ĠSoĠifĠweĠproceed...ĊĊButĠthisĠisĠconsistentĠwithĠourĠcalculations.ĠSoĠyes,ĠthenĠtheĠeighthĠterm
Rollout 11, Act: 2.005
Ġbc+Ġca.ĠcomputeĠp2.ĠMaybeĠthatĠexpandsĠtoĠsomethingĠthatĠcanĠbeĠcombinedĠwithĠq,Ġbut
Rollout 1, Act: 1.965
ĠequationsĠintersectĠatĠaĠcommonĠy.ĊĊButĠtheĠproblemĠnowĠisĠtoĠshowĠthatĠtheĠsystemĠofĠequationsĠ||x
Rollout 46, Act: 1.962
ĠinĠtheĠrotatingĠframe.ĠAlternatively,ĠmaybeĠit'sĠfromĠaĠLagrangianĠwhereĠtheĠvariablesĠareĠtheĠrotating
Rollout 91, Act: 1.957
Ġ)Ġ=Ġ(ĠtĠ-1Ġ,ĠtĠ)ĊĊSoĠnow,ĠweĠcanĠcomputeĠtheĠanglesĠatĠBĠandĠC
Top 16 Negative Activations
Rollout 83, Act: -1.957
ĠHence,ĠtheĠlabelsĠmustĠformĠcyclesĠunderĠrotationĠofĠkĠpositions.ĊĊTherefore,ĠtheĠcycleĠlengthĠforĠaĠrotation
Rollout 15, Act: -1.959
Ġk,ĠthenĠanyĠinitialĠstateĠwillĠreachĠzeroĠafterĠkĠsteps.ĊĊTherefore,ĠifĠweĠcanĠshowĠthatĠM
Rollout 83, Act: -1.976
ĠcyclesĠofĠlengthĠ9/d.ĠSoĠrotationsĠbyĠkĠwhereĠgcd(9,k)=dĠwillĠhaveĠcycle
Rollout 83, Act: -1.978
ButĠeachĠgroupĠmustĠhaveĠthreeĠelementsĠassignedĠtheĠsameĠresidue.ĠRemember,ĠresidueĠcounts:Ġ0,1
Rollout 93, Act: -1.986
1007ĠforĠj=1007.ĊĊSimilarly,Ġj=2014:
Rollout 16, Act: -1.988
ĠEachĠsaplingĠindependentlyĠsucceedsĠwithĠprobabilityĠ1/2ĠeachĠday.ĠSoĠtheĠprocessĠcontinuesĠuntilĠallĠsix
Rollout 93, Act: -1.989
Ġchoose1007ĠtimesĠ1007Ġover2^{2014}},Ġplus2
Rollout 83, Act: -1.990
ĠWaitĠno:ĠresidueĠgroupsĠareĠassignmentsĠofĠdigitsĠintoĠresidueĠclasses,ĠwhichĠareĠfixed.ĠSoĠevenĠifĠa
Rollout 56, Act: -1.992
ĠKĠsuchĠthatĠherĠpastureĠneverĠexceedsĠanĠareaĠofĠK".ĊĊGrammatically:ĠafterĠeachĠextension,Ġ"
Rollout 16, Act: -1.994
ĠThenĠtheĠprobabilityĠthatĠitĠhasn'tĠbeenĠconvertedĠafterĠtĠdaysĠisĠ(1/2)^t.ĠTherefore
Rollout 14, Act: -2.003
ĠtheĠmaximumĠpossibleĠratioĠisĠapproximatelyĠ2.25/8âīĪ0.28.ĠWait,
Rollout 83, Act: -2.034
ĠresiduesĠneeded:ĊĊGroup0ĠhasĠthreeĠpositions,ĠassignedĠresidueĠaĠ-->ĠsoĠthat'sĠthreeĠinstancesĠofĠaĠresidue
Rollout 93, Act: -2.034
ĠaĠrootĠofĠunityĠofĠorderĠ1007/(k,1007).ĠWait,
Rollout 16, Act: -2.045
Ġ(sapling)ĠindependentlyĠwithĠprobabilityĠ1/2.ĊĊButĠisĠthereĠanĠanalogousĠformulaĠhere?ĠLet
Rollout 65, Act: -2.118
OriginalĠdefinition:ĠsinceĠcĠisĠderivedĠfromĠaĠviaĠc=19952/Ġa.ĠIf
Rollout 93, Act: -2.154
007ĠimpliesĠjĠâī¡1007ĠmodĠ1007,Ġi.e.,

DOWN_PROJ

Top 16 Positive Activations
Rollout 98, Act: 2.577
,ĠRGBĠstarĠ(1.5Msun)**ĠĠĊĠĠĠ-Ġ**Invalid**ĠĠĊĠĠĠ-Ġ**Population
Rollout 27, Act: 2.453
/2))^(2/3)ĊĊwhereĠÎ(3/2)ĠisĠApéry's
Rollout 27, Act: 2.361
§ĠisĠtheĠreducedĠPlanckĠconstant,ĠandĠÎ(3/2)ĠisĠtheĠRiemann
Rollout 3, Act: 2.270
ĠsquareĠspeed,ĠwhichĠisĠrelatedĠtoĠtemperatureĠandĠmolarĠmass.ĠWait,ĠbutĠtheĠquestionĠspecificallyĠsaysĠ"
Rollout 27, Act: 2.254
ĠmĠk_B)Ġ)Ġ*Ġ(1Ġ/ĠÎ(3/2))^(2/3)ĊĊ
Rollout 27, Act: 2.230
ĠSoĠmĠ=Ġ9Ġ*Ġ1.6605e-27ĠâīĪĠ1
Rollout 3, Act: 2.195
ĠtheĠtemperatureĠinĠKelvin,ĠandĠMĠisĠtheĠmolarĠmass.ĠThenĠthere'sĠtheĠaverageĠspeed,Ġwhich
Rollout 27, Act: 2.173
Ġunits).Ġ1ĠuĠ=Ġ1.6605e-27Ġkg.ĠSoĠm
Rollout 27, Act: 2.131
Ġ\text{u}Ġ=Ġ1.6605Ġ\timesĠ10^{-27
Rollout 17, Act: 2.129
,ĠwhereĠ$a_{0}=5.29Ġ\timesĠ10^{-11}Ġ\
Rollout 22, Act: 2.127
ĠRĠisĠtheĠgasĠconstantĠandĠMĠisĠtheĠmolarĠmass.ĠWait,Ġreally?ĠSoĠthere'sĠanother
Rollout 22, Act: 2.126
P)Ċ-Ġ\(ĠMĠ\)ĠisĠtheĠmolarĠmassĠofĠhydrogenĠgasĠ(\(Ġ0.0
Rollout 3, Act: 2.078
ĠKelvin,Ċ-Ġ\(ĠMĠ\)ĠisĠtheĠmolarĠmassĠofĠradonĠinĠkg/mol.ĊĊ**Steps
Rollout 22, Act: 2.070
ĠinĠgrams,ĠconfusedĠwithĠkg.ĊĊWaitĠtheĠmolarĠmassĠisĠ2.016Ġg/mol
Rollout 93, Act: 2.066
ĠconnectedĠtoĠsum=Ġ2^{2nĠ-1}Ġ(1Ġ+ĠC(2n,n)
Rollout 22, Act: 2.043
,ĠeachĠmoleculeĠhasĠtwoĠhydrogenĠatoms.ĠTheĠmolarĠmassĠofĠhydrogenĠisĠapproximatelyĠ2.01
Top 16 Negative Activations
Rollout 95, Act: -1.894
3*(k/2450),Ġsqrt(671)/3*(k/24
Rollout 95, Act: -1.894
ĠupĠtoĠ(25/3Ġa,Ġsqrt(671)/3ĠaĠ),ĠadndĠlength
Rollout 95, Act: -1.899
3)k/2450,Ġsqrt(671)/3Ġ*k/24
Rollout 93, Act: -1.900
2...ĠSoĠevenĠtheĠfirstĠtermĠwouldn'tĠdecayĠthatĠmuch?ĠWait,ĠbutĠifĠthoseĠapproximationsĠapply
Rollout 41, Act: -1.902
.ĠFromĠthis,ĠsubtractĠtheĠ9ĠnumbersĠwithĠrepeatingĠdigits.ĠSoĠ90Ġ-Ġ9Ġ=
Rollout 95, Act: -1.910
450)*(-2/3,Ġsqrt(671)/3),ĠwhichĠisĠinĠthe
Rollout 42, Act: -1.910
ĠcentralĠstructureĠis:ĊĊCyclobutyl-C(OH)H-CyclopropylĊĊBut
Rollout 38, Act: -1.920
Ġ[31]2Ġ(x2Ġ+ĠW2)ĊĊComputeĠleftĠside:ĊĊ625(WĠ+
Rollout 95, Act: -1.927
)ĠtoĠC(25/3,Ġsqrt(671)/3),ĊĊĠSoĠAQkĠis
Rollout 38, Act: -1.933
Ġ=Ġ6.2Ġsqrt(x2Ġ+ĠW2)ĊĊLetĠmeĠexpressĠ6.2ĠasĠ
Rollout 9, Act: -1.938
ĠandĠGM.ĊĊEquationĠforĠGEĠ(fromĠG(10Ġ-Ġh,5)ĠtoĠE(
Rollout 38, Act: -1.959
2Ġ=Ġ961(x2Ġ+ĠW2)ĊĊDivideĠbothĠsidesĠbyĠ(WĠ+Ġx
Rollout 1, Act: -1.971
ĠdĠ/Ġsqrt(2)Ġ+ĠdĠ/Ġsqrt(2)Ġ=ĠdĠsqrt(2}Ġ~1
Rollout 95, Act: -1.973
90*(c/sqrt(3))*sqrt(3Ġ)=490*c,åħ¶ä¸Ńc
Rollout 71, Act: -1.994
d,ĠandĠthenĠ3d2Ġ+Ġe2=2011.ĠThenĠk=2
Rollout 3, Act: -2.151
ĠtheĠmeanĠspeedĠisĠcalledĠaverageĠspeedĠandĠgivenĠbyĠthatĠformulaĠwithĠ8/ÏĢ,ĠwhileĠtheĠrms

Layer 15

GATE_PROJ

Top 16 Positive Activations
Rollout 45, Act: 1.439
ĠthatĠĠ$Ġf(1)Ġ\equal{}Ġ1,Ġf(2)Ġ\equal{}Ġ
Rollout 49, Act: 1.404
ĠsubsetĠproperty?ĠIĠthinkĠso.ĠLetĠmeĠcheckĠthis.ĊĊIfĠXĠisĠhereditarilyĠLindelö
Rollout 1, Act: 1.368
ĠareĠmutuallyĠorthogonalĠafterĠscaling.ĊĊWait,ĠletĠmeĠrecallĠthatĠifĠyouĠhaveĠaĠregularĠsimplexĠinĠR^n
Rollout 9, Act: 1.358
}ĊĊWait,ĠthatĠseemsĠcorrect.ĊĊLetĠmeĠcheckĠthis.ĊĊGivenĠthatĠtheĠareaĠisĠ100
Rollout 88, Act: 1.354
ĠreferringĠtoĠweights.ĠHmm.ĊĊWait,ĠletĠmeĠcheckĠagain:Ġ"WhatĠisĠtheĠtotalĠnumberĠofĠweights
Rollout 33, Act: 1.328
Ġ(6Ġ-Ġkm)!.ĊĊWait,ĠletĠmeĠcheckĠwithĠaĠconcreteĠexample.ĠTakeĠA_1ĠâĪ
Rollout 5, Act: 1.326
ĠtheĠtwoĠbases.ĠWait,Ġno,ĠletĠmeĠcheck.ĠĊĊIĠrememberĠthatĠforĠareasĠdividedĠbyĠa
Rollout 39, Act: 1.263
Ġtotal?ĠWait,Ġno.ĠWait,ĠletĠmeĠcheck.ĠIfĠtheĠfirstĠtermĠisĠ1Ġdigit,
Rollout 18, Act: 1.258
ĠneedĠxĠtoĠbeĠpositive.ĊĊFirst,ĠletĠmeĠrecallĠthatĠinĠZ2,ĠanyĠsubgroupĠisĠisomorphic
Rollout 15, Act: 1.257
ĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$n\geq1$ĠĠbeĠaĠpositiveĠinteger.
Rollout 90, Act: 1.249
ĠitselfĠisĠ4Ġdigits).ĠWait,ĠletĠmeĠthinkĠagain.ĊĊWait,ĠholdĠon.Ġ20
Rollout 45, Act: 1.242
qĠmx^2\plus{}(5m\plus{}1)x\plus{}4m$
Rollout 1, Act: 1.234
ĠtwoĠdistinctĠvectorsĠareĠorthogonal.ĊĊFirst,ĠletĠmeĠrecallĠthatĠanĠorthonormalĠsystemĠrequiresĠtwoĠthings:
Rollout 74, Act: 1.233
ĠtoĠtheĠmagneticĠfieldsĠcausingĠtheĠdeviation.ĠLetĠmeĠcheckĠmyĠnotes.ĠOh,Ġright!ĠInĠtheĠcontext
Rollout 65, Act: 1.214
Ġc)ĠsatisfyĠthisĠcondition.ĊĊFirst,ĠletĠmeĠrecallĠwhatĠsimilarĠprismsĠmean.ĠIfĠoneĠprismĠis
Rollout 66, Act: 1.213
ĠtoĠfindĠtheĠperimeterĠp.ĊĊAlright,ĠletĠmeĠrecallĠthatĠinĠaĠright-angledĠtriangle,ĠtheĠaltitude
Top 16 Negative Activations
Rollout 42, Act: -3.537
Ġring.ĠButĠcyclopropaneĠisĠthree-membered;ĠformingĠaĠdoubleĠbondĠadjacentĠtoĠitĠwouldĠcreate
Rollout 2, Act: -3.548
00ĊĊSimilarly,10^kĠ-1=Ġ-1Ġmod1000Ġfork>=
Rollout 42, Act: -3.559
propane)ĠandĠtheĠotherĠisĠ4-memberedĠ(cyclobutane),ĠlinkedĠthroughĠa
Rollout 70, Act: -3.564
^4Ġ-1=2400-1=2400)ĠbeĠaĠ7
Rollout 69, Act: -3.586
ometricĠidentity.Ġ2ĠcosĠ2ĠthetaĠ-1ĠisĠequalĠtoĠcosĠ2Ġtheta?ĠWait,
Rollout 42, Act: -3.591
ĠaĠdoubleĠbondĠsuchĠthatĠoneĠringĠremainsĠthree-memberedĠandĠtheĠotherĠpartĠisĠaĠfour-membered.ĊĊ
Rollout 64, Act: -3.611
Ġ)ĊĊSinceĠ2^{2009}-1ĠisĠlarge,ĠtheĠfractionĠisĠjustĠunderĠ8
Rollout 15, Act: -3.612
ĠBecauseĠn=1ĠisĠ2^1Ġ-1Ġ=1,ĠandĠthenĠmaybeĠ3,Ġ
Rollout 64, Act: -3.613
ĠButĠ2^{2012}Ġ-1Ġ-KĠyĠisĠalreadyĠanĠintegerĠbecauseĠbothĠM
Rollout 64, Act: -3.627
ĠwhileĠ2^{2009}Ġ-1ĠisĠpositiveĠforĠa2.ĊĊSoĠweĠcanĠthink
Rollout 42, Act: -3.629
Ġring.ĠCyclopropaneĠisĠ3-membered;ĠopeningĠcouldĠcreateĠaĠ4-,Ġ5
Rollout 42, Act: -3.629
ĠringsĠandĠaĠcentralĠcarbon,ĠgettingĠaĠsix-memberedĠringĠseemsĠunclear.ĠOptionĠCĠisĠhexahy
Rollout 15, Act: -3.632
=3,ĠwhichĠisĠ2^2Ġ-1:Ġno,ĠsinceĠn=3ĠwasĠbad.
Rollout 42, Act: -3.637
ĠspiroĠunionĠwhereĠoneĠringĠisĠ3-memberedĠ(cyclopropane)ĠandĠtheĠother
Rollout 2, Act: -3.648
;ĊĊk=2,10^2Ġ-1=99ĊĊk=3,Ġso1
Rollout 64, Act: -3.659
SinceĠ2^{2009}Ġ-1Ġ=ĠaĠveryĠlargeĠnumber,Ġ7/(2

UP_PROJ

Top 16 Positive Activations
Rollout 77, Act: 2.110
ĠCalculateĠ$f(14,52)$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 90, Act: 2.109
ĠareĠdivisibleĠbyĠĠ$2018$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI
Rollout 11, Act: 2.096
ĠvaluesĠofĠtheĠorderedĠpairĠĠ$(p,q)$Ġ.ĊĊ*DavidĠAltizio*Ċ<|im_start|>assistantĊ<|im_start|>think
Rollout 39, Act: 2.093
ĠFindĠtheĠsumĠofĠtheĠdigitsĠofĠ$N$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠme
Rollout 78, Act: 2.084
ĠrelativelyĠprimeĠpositiveĠintegers.ĠFindĠ$m+n$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠme
Rollout 84, Act: 2.082
$ĠasĠmuchĠbeverage.ĠFindĠ$m+n$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 9, Act: 2.054
ĠtoĠ$EM$ĠinĠ$\triangleĠGEM$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 59, Act: 2.049
ĠprimeĠpositiveĠintegers.ĠFindĠ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠthe
Rollout 70, Act: 2.041
-10ĠnumberĠthatĠisĠtwiceĠ$N$Ġ.ĠForĠexample,Ġ$51$ĠisĠaĠ
Rollout 5, Act: 2.008
ĠexceedĠ$x^2/100$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet's
Rollout 16, Act: 2.007
Ġ.ĠFindĠĠ$100m+n$Ġ.ĊĊ*ProposedĠbyĠYannickĠYao*Ċ<|im_start|>
Rollout 8, Act: 1.991
ĠprimeĠpositiveĠintegers.ĠFindĠ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet's
Rollout 50, Act: 1.986
ĠrelativelyĠprimeĠpositiveĠintegers.ĠFindĠ$m+n$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠme
Rollout 20, Act: 1.946
ĠfactorsĠinĠtheĠprimeĠfactorizationĠofĠ$n$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 15, Act: 1.934
ĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$n\geq1$ĠĠbeĠaĠpositive
Rollout 45, Act: 1.934
ĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$Ġf:ĠZĠ\toĠZ$ĠĠbeĠsuch
Top 16 Negative Activations
Rollout 44, Act: -1.913
ĠtemperatureĠchanges.ĠBecauseĠinĠtheĠexampleĠwhereĠIĠtookĠreversibleĠisothermal,ĠbutĠifĠinsteadĠtheĠprocessĠisĠreversible
Rollout 11, Act: -1.917
1.ĠThen,ĠpĠ=ĠxĠ+ĠyĠ+ĠzĠ+ĠxyĠ+ĠyzĠ+Ġzx.ĠAndĠq
Rollout 11, Act: -1.925
Ġr_aĠe^{iθ_a},ĠbĠ=Ġr_bĠe^{iθ_b},ĠcĠ=Ġr
Rollout 11, Act: -1.926
ĠforĠsomeĠintegerĠk.ĠHmm,ĠasĠmodulus:Ġr_aĠr_bĠr_cĠ=Ġ1.ĊĊBut
Rollout 11, Act: -1.933
¸,ĠbĠ=ĠaĠ(soĠsameĠmodulusĠandĠargument),ĠcĠ=1/a2.ĠThenĠcomputeĠp
Rollout 11, Act: -1.935
ĠconsiderĠallĠthreeĠvariablesĠwithĠtheĠsameĠmodulus,ĠsayĠr,ĠbutĠdifferentĠarguments.ĠThenĠsinceĠproductĠisĠ
Rollout 69, Act: -1.948
ĠatĠtheĠfirstĠterm:ĊĊInĠterm_0,ĠangleĠ=Ġ10°,ĠsoĠsin2(
Rollout 69, Act: -1.950
(10°)ĊĊInĠterm_2,Ġangle=10°/9:Ġsin2(
Rollout 11, Act: -1.955
Ġ+1/r3.ĊĊNoteĠthatĠifĠRĠ=Ġr3Ġ+1/r3,ĠthenĠcosθĠ=
Rollout 69, Act: -1.968
iâĪĴ1})ĊĊLetĠmeĠdenoteĠthatĠphiĠ=Ġtheta_{iâĪĴ1}ĊĊTherefore,Ġterm_iĠ=
Rollout 11, Act: -1.995
ĠsuchĠthatĠaĠ=Ġ\overline{b},ĠcĠisĠreal?ĠButĠthenĠcĠmustĠnotĠbeĠreal
Rollout 37, Act: -2.056
,2,...,2015,ĠandĠk_j=ĠjĠforĠsimplicity.ĠThenĠmatrixĠAĠis
Rollout 91, Act: -2.083
ĠD.ĠSinceĠBDĠisĠequalĠtoĠDC,ĠbutĠanglesĠoppositeĠtoĠthemĠmightĠnotĠbeĠequal.ĠWait,
Rollout 69, Act: -2.095
=0ĠtoĠinfinity,ĠwhichĠcorrespondsĠtoĠtermsĠwithĠanglesĠ10°,Ġ10/3
Rollout 11, Act: -2.117
1.ĠSinceĠ1ĠhasĠmodulusĠ1ĠandĠargumentĠ0,ĠsoĠmodulusĠr_aĠr_bĠr
Rollout 69, Act: -2.138
(30°)ĊĊInĠterm_1,ĠangleĠ=10°/3:Ġsin2(

DOWN_PROJ

Top 16 Positive Activations
Rollout 64, Act: 2.043
)*2^nĠ=Ġk*2^{n-1},ĠwhichĠisĠintegerĠforĠnĠâī¥1.ĠTherefore
Rollout 17, Act: 2.036
θ.ĊĊLetĠmeĠcomputeĠtheĠpolarĠintegral.ĠLetĠuĠ=Ġcosθ,ĠduĠ=Ġ-sinθĠd
Rollout 87, Act: 2.031
ĠtoĠstandardĠintegrals:ĊĊTheĠintegralĠâĪ«ĠduĠ/Ġsqrt(u2Ġ+Ġc)Ġ=ĠsinhâĪĴ1
Rollout 87, Act: 2.018
:ĊĊTheĠintegralĠâĪ«0^aĠdxĠ/Ġsqrt(x2Ġ+Ġy2Ġ+Ġz2)Ġ=
Rollout 49, Act: 2.013
ĠspacesĠthatĠareĠnotĠregular,ĠcalledĠTychonoffĠcorkscrewĠorĠsimilarĠexamples.ĠButĠifĠthe
Rollout 87, Act: 2.006
Ġintegral,ĠIĠcanĠuseĠsubstitution.ĠLetâĢĻsĠsetĠuĠ=Ġx,ĠsoĠtheĠintegrandĠisĠ1
Rollout 73, Act: 2.004
-Ġ(n+1)2^{n+1}Ġ+Ġn2^{n+2}Ġ=
Rollout 87, Act: 1.999
ĠstandardĠintegrals:ĊĊTheĠintegralĠâĪ«ĠduĠ/Ġsqrt(u2Ġ+Ġc)Ġ=ĠsinhâĪĴ1(u
Rollout 87, Act: 1.963
«ĠrĠlnĠrĠdr.ĠLetâĢĻsĠdoĠsubstitutionĠuĠ=ĠlnĠr,ĠdvĠ=ĠrĠdr.ĠThen
Rollout 51, Act: 1.947
\Ġdx.Ġ\]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠintegralĠtoĠsolve:
Rollout 44, Act: 1.943
20*(20^(1/3))^2âīĪ20*(2.714)^
Rollout 87, Act: 1.922
.ĊĊPerhapsĠinsteadĠdoĠaĠsubstitution.ĠLetâĢĻsĠsetĠuĠ=Ġy,ĠvĠ=Ġz.ĠNotĠhelpful.
Rollout 86, Act: 1.904
ĠwithĠxĠsuchĠasĠx*sinx*cosx^2.ĠThatĠwasĠcase9âīĪ0.22
Rollout 87, Act: 1.881
Ġintegrals:ĊĊTheĠintegralĠâĪ«ĠduĠ/Ġsqrt(u2Ġ+Ġc)Ġ=ĠsinhâĪĴ1(uĠ/
Rollout 12, Act: 1.875
_a^bĠ\frac{dt}{\sqrt{(tĠ-Ġa)(bĠ-Ġt)(t
Rollout 57, Act: 1.857
.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠifĠwe
Top 16 Negative Activations
Rollout 54, Act: -2.586
ĠconjugateĠbase,ĠethanethiolateĠ(HSâĪĴĠorĠCH2CH2SâĪĴ),Ġis
Rollout 54, Act: -2.637
ĠtoĠworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<ĠHOâĪĴĠ<ĠROâĪĴ.ĠWait
Rollout 54, Act: -2.663
ĠstrongerĠacidsĠthanĠalcohols,ĠsoĠthiolateĠisĠaĠweakerĠbase?ĠWait,ĠpKaĠof
Rollout 54, Act: -2.671
SĠhasĠaĠpKaĠaroundĠ7.ĠSoĠHSâĪĴĠisĠaĠweakerĠbaseĠthanĠOHâĪĴ,Ġright
Rollout 20, Act: -2.672
)^ÃĹĠisĠcyclic,ĠthereĠexistĠprimitiveĠrootsĠwhereĠord_m(p)Ġ=ĠmĠ-1,ĠandĠelements
Rollout 54, Act: -2.737
KaĠ~10ĠofĠethanethiolĠâĨĴĠHS-RĠpKaĠaroundĠ10).ĠWait,
Rollout 54, Act: -2.746
iolate.ĠPropionateĠisĠaĠcarboxylate,ĠRCOOâĪĴ,ĠsoĠeachĠofĠthem
Rollout 21, Act: -2.775
,ĠtheĠfunctionalĠequationĠgivenĠisĠsimilarĠtoĠCauchyĠbutĠincremental.ĠLetâĢĻsĠsee:Ġf(xĠ+
Rollout 54, Act: -2.803
.ĠWait,Ġno.ĠTypically,ĠcarboxylatesĠareĠlessĠnucleophilicĠthanĠhydroxide,
Rollout 54, Act: -2.845
Ġsolution:ĊĊInĠSN2Ġreactions,ĠtheĠthiolateĠionĠ(e.g.,ĠSHâĪĴ)ĠisĠa
Rollout 54, Act: -2.937
ĠofĠH2SĠisĠaroundĠ7,ĠsoĠHSâĪĴĠisĠaĠstrongerĠbaseĠthanĠOHâĪĴ?ĠWait
Rollout 54, Act: -2.953
ĠmoreĠpolarizable.ĠForĠexample,ĠinĠwater,ĠHSâĪĴĠisĠaĠbetterĠnucleophileĠthanĠOHâĪĴĠdespite
Rollout 54, Act: -3.041
ĠpropionateĠ(whichĠitĠisâĢĶcarboxylateĠresonance),ĠthenĠpropionateĠwouldĠhaveĠaĠmuchĠlower
Rollout 54, Act: -3.093
ĠThatĠmakesĠsense.ĊĊEthanethiolateĠ(HSâĪĴĠorĠRSâĪĴ):ĠSulfurĠisĠlarger
Rollout 54, Act: -3.127
Ġworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<ĠHOâĪĴĠ<ĠROâĪĴ.ĠWait,
Rollout 54, Act: -3.470
3OH)Ċ5.ĠEthanethiolateĠ(HSâĪĴĠorĠmaybeĠC2H5SâĪĴ?

Layer 16

GATE_PROJ

Top 16 Positive Activations
Rollout 11, Act: 4.026
ĠtheĠsameĠmodulus.ĊĊAlternatively,ĠtwoĠvariablesĠwithĠreciprocalĠmodulusĠandĠoneĠwithĠmodulusĠnotĠ1.ĊĊButĠoverall
Rollout 11, Act: 3.815
Ġ=Ġ\overline{1/b},ĠleadingĠtoĠmodulusĠ|a|Ġ=1/|bĠ|Ġ;
Rollout 11, Act: 3.686
Ġreal.ĠButĠsinceĠaĠisĠneitherĠrealĠnorĠofĠmodulusĠ1,ĠthisĠisĠnotĠtheĠcase.ĊĊWait
Rollout 11, Act: 3.657
|=1/k2Ġ<1.ĠSoĠnoneĠhaveĠmodulusĠ1,ĠwhichĠisĠokay.ĠThen,Ġset
Rollout 11, Act: 3.632
Ġ=Ġθ,ĠbĠ=ĠaĠ(soĠsameĠmodulusĠandĠargument),ĠcĠ=1/a2.ĠThen
Rollout 11, Act: 3.610
Ġ1.ĠLetâĢĻsĠconsiderĠthemĠwithĠmoduliĠreciprocal-related.ĠLikeĠ|a|=k>1,
Rollout 11, Act: 3.574
?ĊĊAlternatively,ĠrecallĠtheĠproblemĠstatesĠthatĠnoneĠhaveĠmodulusĠ1.ĠIfĠweĠletâĢĻsĠtakeĠa,
Rollout 11, Act: 3.572
.ĠHence,ĠallĠthreeĠvariablesĠcannotĠhaveĠtheĠsameĠmodulus.ĊĊAlternatively,ĠtwoĠvariablesĠwithĠreciprocalĠmodulusĠandĠone
Rollout 11, Act: 3.570
1.ĠButĠproblemĠstatesĠthatĠnoneĠofĠthemĠhaveĠmodulusĠ1.ĠHence,Ġ|a|ĠandĠ|
Rollout 11, Act: 3.539
ĠmodulusĠ1.ĠLetâĢĻsĠconsiderĠthemĠwithĠmoduliĠreciprocal-related.ĠLikeĠ|a|=k>1
Rollout 72, Act: 3.521
ĠSo,ĠmaybeĠtheĠsumĠcanĠbeĠarrangedĠtoĠbeĠreal.ĠIndeed,ĠforĠexample,Ġ1Ġ+ĠÎ
Rollout 11, Act: 3.495
ĠproductĠisĠ1.ĠButĠaĠandĠbĠhaveĠmodulusĠrĠâīł1,ĠcĠhasĠmodulusĠ1/(
Rollout 52, Act: 3.484
ĠpositiveĠ2014.ĠLet'sĠcheckĠpartialĠsum.ĊĊButĠinĠreality,ĠtheĠsumĠfromĠ-
Rollout 11, Act: 3.473
},Ġc=Ġe^{-i2θ}.ĠButĠmodulusĠofĠaĠandĠbĠisĠ1,ĠwhichĠis
Rollout 11, Act: 3.451
ĠnoneĠofĠa,Ġb,ĠcĠareĠrealĠorĠmodulusĠ1.ĠLetâĢĻsĠconsiderĠthemĠwithĠmoduli
Rollout 11, Act: 3.428
ĠtheirĠproductĠisĠ1,ĠsoĠifĠtwoĠhaveĠmodulusĠgreaterĠthanĠ1ĠandĠoneĠless,ĠbutĠtheir
Top 16 Negative Activations
Rollout 91, Act: -1.802
2}Ġ=Ġsqrt{2t2Ġ-Ġ2tĠ+1Ġ}.ĊĊSoĠsin(theta_CĠ)
Rollout 36, Act: -1.812
Ġ-Ġ2sxĠ-Ġ2syĠ-Ġ2szĠ+Ġ3s2Ġ=Ġ90
Rollout 36, Act: -1.813
0Ġ-Ġs2Ġ-Ġ2syĠ-Ġ2szĠ+Ġ3s2Ġ=Ġ90
Rollout 66, Act: -1.815
Ġ6abĠ-Ġ6b2Ġ-Ġ6bĠcĠ=Ġ0ĊĊHmm,ĠperhapsĠgroup
Rollout 97, Act: -1.822
1}\right)Ġ=Ġ2Ġ\timesĠ5Ġ\timesĠ10^{-4}Ġ\times
Rollout 28, Act: -1.825
2Ġ/Ġ(2hĠ-a)Ġ-Ġ2ĠhĠb2Ġ/Ġ(2kĠ-b)ĊĊ
Rollout 28, Act: -1.849
Ġ\frac{2AĠa^2Ġ+Ġ2CĠb^2}{ab}Ġ=Ġ2
Rollout 36, Act: -1.862
Ġ-Ġ2sxĠ-Ġ2syĠ-Ġ2szĠ+Ġ3s2Ġ=Ġ90
Rollout 36, Act: -1.865
Ġ-Ġ2sxĠ-Ġ2syĠ-Ġ2szĠ+Ġ3s2Ġ=Ġ90
Rollout 36, Act: -1.872
Ġ-Ġ2sxĠ-Ġ2syĠ-Ġ2szĠ+Ġ3s2Ġ=Ġ90
Rollout 28, Act: -1.874
ĠhkĠ-Ġ2aĠb2ĠhĠ-Ġ2a2ĠbĠkĠ+Ġa2Ġb2Ġ+
Rollout 28, Act: -1.881
ĠhkĠ-Ġ2aĠb2ĠhĠ-Ġ2a2ĠbĠkĠ+Ġa2Ġb2Ġ=
Rollout 28, Act: -1.946
:ĊĊ\(Ġ(-2AĠa^2Ġ-Ġ2CĠb^2)Ġ+ĠBĠabĠ=Ġ
Rollout 36, Act: -1.954
Ġ-Ġs2)Ġ-Ġ2sxĠ-Ġ2syĠ-Ġ2szĠ+Ġ3s2
Rollout 36, Act: -2.019
sxĠ-Ġs2Ġ-Ġ2sxĠ-Ġ2syĠ-Ġ2szĠ+Ġ3s2
Rollout 28, Act: -2.096
Ġa2Ġ(2kĠ-b)Ġ-Ġ2ĠhĠb2Ġ(2hĠ-aĠ)ĊĊExpand

UP_PROJ

Top 16 Positive Activations
Rollout 16, Act: 2.163
ĠeachĠremainingĠsaplingĠhasĠaĠ1/2ĠchanceĠtoĠconvert.ĠLetĠmeĠmodelĠthisĠasĠaĠMark
Rollout 36, Act: 2.141
:ĊĊButĠIĠneedĠtoĠbeĠcareful.ĠSoĠtheĠcomputeĠshowedĠthatĠS=36,864
Rollout 16, Act: 2.115
ĠandĠeachĠsaplingĠhasĠaĠ1/2ĠchanceĠofĠturningĠintoĠaĠperfectĠtreeĠthatĠday.ĠOnce
Rollout 89, Act: 2.096
.ĊĊButĠunlessĠspecified.ĠButĠconsideringĠthatĠschoolĠproblemsĠtypicallyĠavoidĠthis.ĠSoĠpossibleĠdupe:ĊĊGirlsĠ
Rollout 16, Act: 2.068
ĠallĠremainingĠsaplingsĠhaveĠaĠ1/2ĠchanceĠofĠconverting.ĠSoĠeachĠday,ĠmultipleĠcanĠbe
Rollout 16, Act: 2.064
ĠstillĠaĠsaplingĠhasĠaĠ1/2ĠchanceĠtoĠconvert.ĠTheĠprocessĠisĠsimilarĠtoĠindependentĠtrials
Rollout 22, Act: 2.042
Ġ"computeĠtheĠmeanĠmolecularĠspeed...",ĠgivenĠthat'sĠunderĠunspecifiedĠtemperature,ĠbothĠareĠreasonable.ĠButĠinĠphysics
Rollout 97, Act: 2.025
âīĪ2.3ãĢĤbutĠgivenĠrealĠexamsĠproblemsĠoftenĠdemandĠwhich,ĠI'mĠtorn.ĊĊTheĠconfusionĠarises
Rollout 39, Act: 2.019
ĠperhapsĠanotherĠexample.ĊĊButĠmaybeĠsteppingĠback.ĠTheĠprincipalĠpartĠofĠtheĠlogicĠisĠthat,ĠgivenĠthatĠeach
Rollout 29, Act: 1.998
ĠexplicitlyĠasksĠforĠtheĠgaugeĠpressure,ĠwhichĠisĠformulaicallyĠPĠ=ĠrhoĠgĠhĠforĠanĠopenĠcontainer,
Rollout 54, Act: 1.962
ethiolate.ĠPropionateĠisĠaĠcarboxylate,ĠRCOOâĪĴ,ĠsoĠeachĠof
Rollout 37, Act: 1.961
ĠwhenĠconsideringĠasĠvariables,ĠtheĠsymmetricĠpolynomialĠSĠisĠsuchĠthatĠn!ĠdividesĠit.ĠWait,ĠbutĠhow
Rollout 64, Act: 1.957
Ġ(2^{3}Ġ-1Ġ)ĊĊWait,ĠlongĠdivision.ĠSinceĠ2012=2
Rollout 25, Act: 1.952
ĠSinceĠgravitationalĠforceĠdependsĠonĠbothĠmassesĠ(Newton'sĠlawĠofĠgravitation),ĠbutĠinĠtheĠcaseĠofĠthe
Rollout 27, Act: 1.951
ĠBeryllium-9ĠhasĠ4ĠprotonsĠandĠ5Ġneutrons,ĠsoĠtotalĠspinĠinteger
Rollout 60, Act: 1.942
.Ċ<|im_start|>userĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequencyĠ$f_{0}=492Ġ\mathrm
Top 16 Negative Activations
Rollout 86, Act: -1.812
Ġ(x2)(1/2)Ġsin2xĠ|Ġfrom0ĠtoĠpi/2Ġ-ĠâĪ«
Rollout 86, Act: -1.827
Ġreturning,ĠtheĠintegralĠâĪ«x2Ġcos2xĠdx=Ġ-âĪ«xĠsin2xĠdx=
Rollout 86, Act: -1.838
ButĠwhenĠweĠcomputeĠâĪ«x2Ġcos2xĠdx=Ġ-ĠâĪ«xĠsin2xĠdx
Rollout 86, Act: -1.841
/4.ĠSoĠâĪ«x2Ġcos2xĠdx=Ġ-pi/4Ġ?ĊĊButĠthatĠcan
Rollout 69, Act: -1.849
Ġtheta)ĊĊ=Ġcos(theta)Ġ-Ġ2Ġcos(theta)Ġsin^2ĠthetaĠ-Ġ2Ġsin^
Rollout 86, Act: -1.855
Ġ+Ġ(1/2)ĠâĪ«cos2xĠdxĠ]ĊĊCalculatingĠthis:ĊĊ-Ġ[Ġ-Ġ(
Rollout 86, Act: -1.858
Ġ+Ġ(1/2)ĠâĪ«Ġcos2xĠdxĠ]ĊĊ=Ġ-Ġ[Ġ-Ġ(1/2
Rollout 86, Act: -1.862
Ġ+ĠC.ĊĊThus,ĠâĪ«sin^3xĠdx=Ġ-cosxĠ+Ġ(1/3
Rollout 86, Act: -1.874
xĠdx,ĠthenĠâĪ«x2Ġcos2xĠdxĠ=Ġ(1/2)x^2Ġsin
Rollout 51, Act: -1.879
Ġ=Ġsin8xĠcos8θĠ+Ġcos8xĠsin8θĊĊTherefore,ĠtheĠintegralĠbecomes:ĊĊ
Rollout 86, Act: -1.881
Therefore,ĠtheĠoriginalĠâĪ«x2Ġcos2xĠdxĠ=Ġpi/4.ĊĊTherefore,ĠreturningĠto
Rollout 86, Act: -1.883
.ĊĊSecondĠintegral:ĠâĪ«x2Ġcos2xĠdx.ĠIntegrationĠbyĠparts.ĊĊLetâĢĻsĠletĠu
Rollout 86, Act: -1.899
Ġ+Ġ(1/2)ĠâĪ«Ġcos2xĠdxĠ)?ĊĊWaitĠno,ĠthatĠearlierĠprocessĠgave
Rollout 86, Act: -1.920
4)ĊĊTherefore,ĠâĪ«x2Ġcos2xĠdxĠ=Ġ-âĪ«xĠsin2xĠdxĠ=
Rollout 86, Act: -1.929
Ġ+Ġ(1/2)ĠâĪ«Ġcos2xĠdx.ĊĊEvaluateĠterms:ĊĊAtĠpi/2:
Rollout 0, Act: -1.933
ĠareĠrelatedĠtoĠtheĠdivisorsĠofĠ20!ĠwhereĠeachĠpairĠconsistsĠofĠtwoĠcoprimeĠnumbers

DOWN_PROJ

Top 16 Positive Activations
Rollout 54, Act: 2.797
ĠbaseĠthanĠhydroxide,ĠbutĠmaybeĠRCOOâĪĴĠisĠmoreĠhindered.ĠWait,ĠbutĠregardless
Rollout 54, Act: 2.629
Ġ(alkoxide)Ġ<ĠHOâĪĴĠ<ĠRCOOâĪĴ.ĠBecauseĠROâĪĴĠisĠstrongestĠbase,Ġmost
Rollout 54, Act: 2.623
ionateĠisĠaĠcarboxylate,ĠRCOOâĪĴ,ĠsoĠeachĠofĠthemĠexceptĠmethanolĠhas
Rollout 54, Act: 2.603
ĠsolventsĠ(fromĠbetterĠtoĠworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<ĠHOâĪĴ
Rollout 54, Act: 2.603
),ĠpropionateĠ(3):ĊĊPropionateĠ(RCOOâĪĴ)ĠisĠtheĠweakestĠbase,ĠsoĠinĠpolar
Rollout 54, Act: 2.586
ĠbeingĠstrongĠbase)ĊĊ3.ĠPropionateĠ(RCOOâĪĴ,ĠresonanceĠstabilized,ĠweakestĠbaseĠamongĠtheĠcharges
Rollout 54, Act: 2.560
ĠpolarĠprotic.ĠSoĠifĠpropionateĠ(RCOOâĪĴ)ĠisĠweakerĠbaseĠthanĠhydroxide,
Rollout 54, Act: 2.522
vated,ĠsoĠworstĠnucleophile.ĠRCOOâĪĴĠisĠweakestĠbase,ĠsoĠleastĠsolvated
Rollout 54, Act: 2.506
ĠOĠnucleophilesĠinĠwaterĠisĠtypicallyĠRCOOâĪĴĠ<ĠHOâĪĴĠ<ĠROâĪĴ.ĠBecauseĠeven
Rollout 12, Act: 2.263
int_1^{\frac{1}{\nu}}Ġ\frac{dx}{\sqrt{(x
Rollout 46, Act: 2.108
ĠangularĠvelocityĠÏī.ĠTheĠaccelerationĠinĠtheĠinertialĠframeĠisĠa_inertialĠ=Ġa_rotating
Rollout 46, Act: 1.995
Ġframe)ĠisĠrelatedĠtoĠtheĠaccelerationĠinĠtheĠinertialĠframeĠaĠby:ĊĊa'Ġ=ĠaĠ-Ġ
Rollout 54, Act: 1.935
ventsĠ(fromĠbetterĠtoĠworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<ĠHOâĪĴĠ<
Rollout 46, Act: 1.926
ĠkineticĠenergyĠisĠdifferent.ĠTheĠvelocityĠinĠtheĠinertialĠframeĠisĠv_inertialĠ=Ġv_rotating
Rollout 54, Act: 1.894
ĠhydroxideĠisĠaĠbetterĠnucleophileĠthanĠacetateĠ(whichĠisĠaĠweakerĠbase).ĠBecauseĠperhapsĠdespite
Rollout 12, Act: 1.886
ĠweĠhadĠxĠfromĠ1ĠtoĠ1/ν,ĠsubstitutionĠt=νxĠgivesĠtĠfromĠν
Top 16 Negative Activations
Rollout 23, Act: -2.264
ĠconsidersĠtheĠalphaĠdecayĠhalfĠlifeĠandĠneglectsĠtheĠbetaĠdecay,ĠtheyĠcalculateĠanswerĠ(A).ĠButĠthe
Rollout 23, Act: -2.269
.Ċ<|im_start|>userĊImagineĠaĠradioactiveĠnucleiĠX(Z,A)ĠcanĠdecayĠintoĠY(Z-2,ĠA
Rollout 23, Act: -2.281
0385ĠsâĪĴ1ĊĊSimilarly,ĠbetaĠdecay:ĠT_beta=0.098
Rollout 23, Act: -2.288
ĠbothĠdecays.ĠSomeoneĠmightĠhaveĠignoreĠtheĠcompetingĠbetaĠdecay.ĠLet'sĠcheckĠthat.ĊĊIfĠthereĠwas
Rollout 23, Act: -2.301
ĠneedĠtoĠconsiderĠtheĠdecayĠconstantsĠforĠbothĠalphaĠandĠbetaĠdecay,ĠsumĠthemĠupĠforĠtheĠtotalĠdecayĠconstant
Rollout 62, Act: -2.305
ĠEuler'sĠfamousĠpolynomialĠn^2Ġ+nĠ+41,ĠwhichĠgeneratesĠprimesĠforĠn=0Ġto
Rollout 27, Act: -2.380
mmm.ĊĊWait,ĠperhapsĠwe'dĠneedĠtheĠidealĠBEĠcondensationĠformula,ĠbutĠagainĠrequiresĠdensity.ĊĊAlternatively
Rollout 23, Act: -2.411
min}^{-1}ĊĠĠĠĠĠ\]ĊĠĠĠ-ĠBetaĠdecayĠhalf-lifeĠ\(ĠT_{1/2,
Rollout 27, Act: -2.495
ĠtheĠBose-EinsteinĠcondensationĠtemperature?ĠBecauseĠtypicalĠBECĠoccursĠatĠnanokelvinĠtoĠmicrokel
Rollout 23, Act: -2.522
ĠdecaysĠintoĠYĠviaĠalphaĠandĠintoĠQĠviaĠBeta,ĠbutĠtheĠactivityĠisn'tĠrelatedĠtoĠbateman
Rollout 27, Act: -2.570
_cĠrelatedĠtoĠwhenĠquantumĠeffectsĠbecomeĠimportant?ĠForĠBEC,ĠT_cĠisĠtheĠcriticalĠtemperatureĠbelowĠwhich
Rollout 23, Act: -2.623
ĠdecayĠandĠ0.098ĠminutesĠforĠbetaĠdecay.ĠLetĠmeĠmakeĠsureĠIĠunderstandĠwhatĠ"
Rollout 23, Act: -2.657
2310ĠminâĪĴ1.ĊĊButĠforĠbetaĠdecay:ĊĊT_beta_1/2=0
Rollout 23, Act: -2.971
ĠcanĠdecayĠinĠtwoĠdifferentĠways:ĠalphaĠdecayĠandĠbeta-minusĠdecay.ĠTheĠquestionĠasksĠforĠtheĠactivityĠof
Rollout 23, Act: -3.043
310ĠminâĪĴ1ĊĊSimilarly,ĠforĠbetaĠdecay,ĠtheĠhalf-lifeĠisĠ0.0
Rollout 23, Act: -3.385
(Z+1,A)ĠbyĠdecayingĠaĠ$\beta^-$ĠwithĠpartialĠhalfĠlifeĠ0.0

Layer 17

GATE_PROJ

Top 16 Positive Activations
Rollout 54, Act: 3.158
ĠForĠexample,ĠinĠanĠSN1Ġreaction,ĠtheĠRDSĠisĠtheĠformationĠofĠcarbocation.ĠIfĠthat
Rollout 54, Act: 3.156
ranceĠmightĠmakeĠitĠlessĠnucleophilicĠbecauseĠtheĠloneĠpairsĠareĠharderĠtoĠaccess,ĠbutĠinĠsolution,
Rollout 54, Act: 3.048
-stabilizedĠanion;ĠtheĠchargeĠisĠdelocalized,ĠwhichĠmakesĠitĠaĠweakerĠnucleophileĠcompared
Rollout 42, Act: 2.996
ĠisĠtypicallyĠ180°,ĠforĠaĠbulkyĠgroupĠinĠaĠstrainedĠenvironment.ĊĊAssumingĠit's
Rollout 54, Act: 2.995
ĠtheĠfollowingĠstatementsĠ(B).ĊĊ1.ĠInĠsubstitutionĠreactionĠifĠtheĠreactantĠinĠrateĠdeterminingĠstepĠisĠcharged
Rollout 54, Act: 2.993
Ġaren'tĠanĠissue.ĠHowever,ĠwithĠstericĠhindrance,ĠbulkyĠstrongĠbasesĠmayĠhaveĠlowerĠnucleophil
Rollout 44, Act: 2.991
1^gamma.ĠSoĠifĠgamma=5/3ĠforĠmonatomicĠgas.ĊĊp1Ġ=Ġp2
Rollout 54, Act: 2.970
Ġhydroxide?ĊĊInĠSN2Ġreactions,ĠbulkinessĠhindersĠtheĠbacksideĠattack,ĠmakingĠitĠless
Rollout 54, Act: 2.941
ĠhydroxideĠisĠbetter.ĊĊButĠwithĠstericĠhindrance:Ġ4-methylcyclohex
Rollout 54, Act: 2.941
ĠproticĠsolventsĠ(fromĠbetterĠtoĠworse):ĠRCOOâĪĴĠ(carboxylate)Ġ<
Rollout 54, Act: 2.938
.ĠPropionateĠisĠaĠcarboxylate,ĠRCOOâĪĴ,ĠsoĠeachĠofĠthemĠexceptĠmeth
Rollout 54, Act: 2.932
ophilicĠinĠwater.ĠWhy?ĠMaybeĠbecauseĠtheĠnegativeĠchargeĠinĠhydroxideĠisĠmoreĠlocalized?ĠSo
Rollout 54, Act: 2.922
.ĊĊSoĠevenĠwithĠstericĠissues,ĠaĠlocalizedĠchargeĠ(likeĠ1)ĠwouldĠmakeĠitĠmoreĠnucle
Rollout 44, Act: 2.921
ĠÎĶUĠ=Ġ-W.ĠForĠadiabaticĠexpansion,ĠWĠisĠpositiveĠ(workĠdoneĠbyĠgas
Rollout 47, Act: 2.902
ĠisĠmovingĠthroughĠaĠtunnel,ĠandĠtheĠcross-sectionalĠareaĠofĠtheĠtrainĠisĠSt,ĠwhileĠtheĠtunnel's
Rollout 54, Act: 2.901
ĠHOâĪĴĠ>ĠROâĪĴ.ĠThisĠisĠbecauseĠofĠpolarizabilityĠandĠsolvation.ĠThiolatesĠ(
Top 16 Negative Activations
Rollout 43, Act: -2.215
>=Ġ(alphaĠ|0>+ĠbetaĠ|1>Ġ+ĠgammaĠ|1>)Ġ\tensorproductĠ|0>Ċ
Rollout 20, Act: -2.215
(an)Ġâī¡Ġ1ĠmodĠ43ĠandĠmodĠ47.ĠToĠfindĠtheĠminimalĠsuch
Rollout 0, Act: -2.218
ĠofĠ20!ĠmustĠgoĠentirelyĠtoĠoneĠofĠtheĠnumbersĠ\(a\)ĠorĠ\(b\).
Rollout 21, Act: -2.223
ĠthatĠfloor(xĠ+Ġn)Ġ=Ġfloor(x)Ġ+nĠforĠintegerĠn.ĠMaybeĠifĠweĠuseĠthat
Rollout 10, Act: -2.228
)ĊĊButĠsinceĠsquaresĠmodulo4ĠareĠ0Ġor1.ĠTherefore,nĠbeing0,1,2,
Rollout 21, Act: -2.233
Ġ=0ĠwouldĠimplyĠthatĠeitherĠf(x)=xĠorĠf(x)=xĠ-1ĠforĠrealĠnumbersĠx
Rollout 10, Act: -2.253
Ġsubtract0Ġor1.ĊĊĠsubtractĠsâī¡0Ġor1Ġmod4,ĠnĠ-sâī¡0Ġ-0
Rollout 21, Act: -2.281
ĠforĠeachĠintegerĠk,ĠfĠmustĠtakeĠvalueĠkĠsomewhere.ĠSinceĠfunctionalĠequationsĠareĠquantifiedĠoverĠallĠvariables
Rollout 20, Act: -2.283
ĠprimeĠp,ĠeitherĠpĠâī¡1ĠmodĠmĠorĠpĠâī¡ĠsomethingĠelse.ĠSoĠcaseĠ1
Rollout 85, Act: -2.299
0,0)$ĠisĠlabeledĠbyĠ0Ġ,ĠandĠforĠeveryĠ$x,ĠyĠ\geqĠ
Rollout 20, Act: -2.302
ĠcasesĠdependingĠonĠwhetherĠpĠâī¡1ĠmodĠmĠorĠnot.ĊĊSoĠforĠmodulusĠmĠ(eitherĠ4
Rollout 43, Act: -2.360
\beta\left|1\right\rangleĠ-\gamma\left|1\right\rangle
Rollout 63, Act: -2.362
Ġ1,ĠqĠ-Ġ1)Ġ\)ĠorĠ\(Ġ(pĠ-Ġ1,ĠqĠ+Ġ
Rollout 64, Act: -2.367
=0.ĠThenĠa_n=Ġ0Ġ+1*2^n=2^n.ĠSoĠitĠworks
Rollout 21, Act: -2.438
Ġf(x)ĠsuchĠthatĠeitherĠf(x)=xĠorĠf(x)=x-1,ĠbutĠthoseĠfunctions
Rollout 10, Act: -2.571
ĠnĠâī¡0Ġmod4:Ġsubtract0Ġor1.ĊĊĠsubtractĠsâī¡0Ġor1Ġmod4,

UP_PROJ

Top 16 Positive Activations
Rollout 80, Act: 2.466
3)ÏĢr3ĠâīĪĠ4/3Ġ*ĠÏĢĠ*Ġ(1.2ĠÃĹĠ1
Rollout 80, Act: 2.382
ĠtheĠvolumeĠofĠtheĠinnerĠcoreĠisĠ(4/3)ÏĢr3.ĠLet'sĠcalculateĠthat:ĊĊ
Rollout 80, Act: 2.336
text{Volume}Ġ=Ġ\frac{4}{3}Ġ\piĠr^3Ġ=Ġ\frac{
Rollout 80, Act: 2.315
0^8Ġcm.ĊĊVolumeĠ=Ġ(4/3)ÏĢr3ĠâīĪĠ4/3
Rollout 25, Act: 2.314
1Ġkm,ĠtheĠvolumeĠwouldĠbeĠ(4/3)ÏĢ*(3,541,0
Rollout 25, Act: 2.311
ĠÏģ,ĠthenĠmassĠmĠ=Ġ(4/3)ÏĢR3Ïģ.ĠSoĠforĠRĠ=
Rollout 81, Act: 2.282
ĠresolveĠthis?ĊĊWait,ĠtangentsĠofĠanglesĠandĠquadrants.ĠIfĠangleĠbetweenĠtwoĠlinesĠcalculatedĠbyĠthe
Rollout 26, Act: 2.238
,ĠbutĠlet'sĠcheckĠtheĠcalculationsĠdecimal-by-decimal.ĊĊÎĶT=14.8ĠK.ĊĊ
Rollout 80, Act: 2.234
)ÏĢ:ĠĊĊVolumeĠâīĪĠ(4/3)Ġ*Ġ3.1416Ġ*
Rollout 99, Act: 2.229
Ġx3,ĠperhapsĠroundingĠto,Ġsay,ĠfourĠdecimalĠplaces.ĠAlternatively,ĠkeepingĠmoreĠdecimals.ĊĊAlternatively,
Rollout 80, Act: 2.210
piĠr^3Ġ=Ġ\frac{4}{3}Ġ\piĠ(1.2Ġ\timesĠ
Rollout 80, Act: 2.203
ĠLet'sĠcalculateĠthat:ĊĊVolumeĠ=Ġ(4/3)ÏĢ(1.2ĠÃĹĠ10
Rollout 69, Act: 2.169
0°=2/âĪļ3âīĪ1.1547...ĊĊTherefore,ĠS=Ġ(1
Rollout 14, Act: 2.146
.31ĠwhichĠcubesĠtoĠ~2.25,ĠwhichĠisĠwayĠsmallĠthan8,ĠindicatingĠthat
Rollout 99, Act: 2.141
ĠperhapsĠduringĠcomputation,ĠweĠneedĠtoĠcarryĠmoreĠdecimalĠplaces.ĊĊLetĠmeĠredoĠtheĠfirstĠiterationĠwithĠmoreĠexact
Rollout 26, Act: 2.118
ĠotherĠvaluesĠ(HgĠenthalpyĠandĠcalorimeterĠcapacityĠareĠgivenĠtoĠtwoĠdecimalĠplaces),ĠlikelyĠthe
Top 16 Negative Activations
Rollout 53, Act: -1.913
2Ġsqrt73Ġ)Ġ=Ġ(73Ġ-27Ġ)Ġ/Ġ(2Ġsqrt73Ġ)=
Rollout 93, Act: -1.915
2:ĊĊ(2*1007Ġ-2j)/1007Ġ=Ġ2(
Rollout 52, Act: -1.926
2014*(2015Ġ-2013)/2=2014
Rollout 38, Act: -1.927
552-2*1055Ġ+1=1,113,02
Rollout 73, Act: -1.938
jĠ>Ġi}Ġ(2^jĠ-Ġ2^i).ĠOurĠgoalĠisĠtoĠcomputeĠthisĠsum
Rollout 67, Act: -1.938
3(nâĢĵ1))*(-180Ġ+24(nâĢĵ1)).ĠLet'sĠcheckĠforĠn
Rollout 64, Act: -1.941
ĠofĠ(2^{2009}Ġ-2):ĊĊ=Ġ\(Ġ(a2)(2^{2
Rollout 18, Act: -1.948
Ġ(Ġ-1Ġ-4Ġy),Ġ(4Ġ-5Ġy).ĠButĠsinceĠxĠdividesĠallĠthreeĠdifferences,
Rollout 70, Act: -1.957
Ġ+Ġ...Ġ+Ġd_1*(10Ġ-2*7)Ġ+Ġd_0*(1Ġ-
Rollout 93, Act: -1.958
Ġexponent:ĊĊFirst,Ġ2014Ġ-2j:Ġ2014=2*
Rollout 93, Act: -1.976
âĢĻ}).ĠButĠe^{i2piĠmâĢĻ}=1ĠifĠmâĢĻĠisĠinteger,ĠsoĠnumeratorĠbecomes
Rollout 56, Act: -2.014
.ĠForĠnâī¥2,Ġ2^nĠ-2ĠincreasesĠrapidly.ĠHere,ĠsequenceĠofĠdenominators:
Rollout 64, Act: -2.029
ĠAĠisĠintegerĠandĠk*2^{nĠ-1}ĠisĠintegerĠforĠnâī¥1,Ġa_n
Rollout 73, Act: -2.104
Ġi,ĠweĠhaveĠsumĠ(2^jĠ-2^i)Ġ=ĠsumĠ2^jĠ-
Rollout 73, Act: -2.104
^{j-1}Ġ(2^jĠ-2^i).ĠButĠourĠsumĠbecameĠN=1
Rollout 73, Act: -2.188
ĠanotherĠway.ĊĊEachĠdifferenceĠ2^jĠ-2^iĠforĠjĠ>Ġi.ĠSoĠmaybe,

DOWN_PROJ

Top 16 Positive Activations
Rollout 98, Act: 2.275
ralĠtypeĠandĠmass:ĠForĠexample,ĠaĠB0ĠstarĠisĠaroundĠ20Msun,ĠB
Rollout 48, Act: 2.238
ĠhexadecimalĠvalue)ĠinĠtheĠIPĠheader:Ġ4500Ġ0034ĠB61
Rollout 23, Act: 2.189
-2,ĠA-4)ĠbyĠemittingĠanĠalphaĠparticleĠwithĠpartialĠhalfĠlifeĠ3.0Ġminutes
Rollout 25, Act: 2.127
-bodyĠproblem,ĠtheĠtrajectoryĠisĠdeterminedĠbyĠtheĠgravitationalĠparameterĠofĠtheĠcentralĠbodyĠ(Earth,ĠinĠthisĠcase
Rollout 46, Act: 2.097
Ġ$U$,ĠwhereĠ$U=-\frac{GĠM_{1}}{\rho_{1}}-\
Rollout 98, Act: 2.092
Ġhelp.ĠAlternatively,ĠIĠcanĠrecallĠthatĠO,ĠB,ĠAĠareĠmassive.ĠLetĠmeĠcheck:ĠB
Rollout 23, Act: 2.081
ĠBatemanĠequationĠifĠXĠdecaysĠintoĠYĠviaĠalphaĠandĠintoĠQĠviaĠBeta,ĠbutĠtheĠactivityĠisn
Rollout 25, Act: 2.073
Ġ-Ġμ/(r),ĠwhereĠμĠisĠtheĠgravitationalĠparameterĠofĠtheĠprimaryĠbodyĠ(Earth,ĠinĠthisĠcase
Rollout 98, Act: 2.071
WDĠ(B5ĠwhenĠinĠtheĠMS)ĠandĠA0VĊĊG2V,ĠK1V
Rollout 25, Act: 2.039
ĠtheĠspecificĠangularĠmomentum,ĠandĠμĠisĠtheĠgravitationalĠparameterĠ(G*M_earth).ĊĊButĠifĠweĠconsider
Rollout 48, Act: 2.034
ĠĠĠ-Ġ**VersionĠandĠIHL:**Ġ`45`ĠinĠhex.ĠVersionĠisĠ4Ġ(IPv
Rollout 22, Act: 2.033
RT/(ÏĢM)),ĠwhereĠRĠisĠtheĠgasĠconstantĠandĠMĠisĠtheĠmolarĠmass.ĠWait,
Rollout 25, Act: 2.029
6.674ÃĹ10^-11ĠNĠm^2/kg^2,ĠmĠis
Rollout 22, Act: 2.025
Ġ(ÏĢĠ*ĠM)).ĊĊRĠisĠtheĠgasĠconstant,ĠwhichĠisĠ8.3144
Rollout 87, Act: 2.011
ĠofĠaĠcubeĠwhereĠtheĠcubeĠhasĠaĠuniformĠchargeĠdensityĠrho,ĠandĠeachĠsideĠisĠlengthĠa.ĠHmm
Rollout 3, Act: 2.002
(3RT/M),ĠwhereĠRĠisĠtheĠgasĠconstant,ĠTĠisĠtheĠtemperatureĠinĠKelvin,ĠandĠM
Top 16 Negative Activations
Rollout 10, Act: -2.398
ĠTheĠkeyĠisĠtoĠidentifyĠlosingĠpositionsĠ(P-positions)ĠwhereĠeveryĠmoveĠleadsĠtoĠaĠwinningĠpositionĠ(
Rollout 42, Act: -2.408
Ġcarbocation.ĠWait,ĠbutĠifĠtheĠmechanismĠisĠE1,ĠthenĠweĠfirstĠgetĠcarbocationĠformation.
Rollout 71, Act: -2.410
8041Ġ-12d2Ġ-12d.ĠNow,Ġ8041
Rollout 10, Act: -2.431
ĠweĠneedĠtoĠfindĠanĠinfiniteĠfamilyĠofĠP-positionsĠ(GrundyĠnumber0).ĠSince,ĠevenĠthough
Rollout 42, Act: -2.449
,ĠalcoholsĠtypicallyĠundergoĠeliminationĠreactions,ĠusuallyĠE1ĠorĠE2ĠmechanismsĠleadingĠtoĠtheĠformationĠof
Rollout 10, Act: -2.461
Ġn=2ĠisĠaĠlosingĠposition,ĠsoĠP-position.ĊĊWait,ĠcontradictsĠmyĠpreviousĠthought.ĠWait
Rollout 10, Act: -2.469
),ĠpositionsĠcanĠoftenĠbeĠclassifiedĠasĠeitherĠN-positionsĠ(nextĠplayerĠcanĠforceĠaĠwin)ĠorĠP
Rollout 38, Act: -2.503
42857.ĠLet'sĠdivideĠnumeratorĠandĠdenominatorĠbyĠ48:Ġ230
Rollout 59, Act: -2.523
6/784.ĊĊFirst,ĠdivideĠnumeratorĠandĠdenominatorĠbyĠ2:Ġ123/
Rollout 90, Act: -2.540
.Ġ112,000Ġ-100,900=11,
Rollout 10, Act: -2.547
oticĠdensityĠzero,ĠthenĠtheĠsetĠofĠP-positionsĠcanĠstillĠbeĠinfinite.ĠTheĠintuitionĠbeingĠthatĠwhen
Rollout 96, Act: -2.569
.ĊĊWait,ĠletĠmeĠcomputeĠthat.ĠDivideĠnumeratorĠandĠdenominatorĠbyĠcommonĠfactorsĠifĠpossible.Ġ17
Rollout 14, Act: -2.689
aĠ-2)}.ĊĊTakeĠreciprocalsĠonĠbothĠsides:ĊĊJ^{1/(2aĠ-1
Rollout 10, Act: -2.695
nextĠplayerĠcanĠforceĠaĠwin)ĠorĠP-positionsĠ(previousĠplayerĠcanĠforceĠaĠwin).ĠTheĠkey
Rollout 10, Act: -2.753
Ġ10,ĠtheĠlosingĠpositionsĠ(P-positions)ĠareĠ0,Ġ2,Ġ5
Rollout 10, Act: -3.224
Ġvictory,ĠthoseĠareĠcalledĠlosingĠpositionsĠorĠP-positionsĠinĠcombinatorialĠgameĠtheory.ĠTheĠproblemĠstates

Layer 18

GATE_PROJ

Top 16 Positive Activations
Rollout 31, Act: 3.531
+1}Ġ+ĠyĠa_nĠforĠnĠâī¥Ġ0,ĠwhereĠxĠandĠyĠareĠrealĠnumbers.
Rollout 68, Act: 3.056
ĠhasĠtoĠbeĠpositive.ĠSimilarly,Ġlog(xĠ+Ġ2)ĠisĠdefinedĠonlyĠwhenĠxĠ+Ġ2
Rollout 47, Act: 3.022
ĠSt,ĠwhileĠtheĠtunnel'sĠisĠS0Ġ=Ġ4St.ĠThatĠmeansĠtheĠareaĠaroundĠtheĠtrain
Rollout 32, Act: 3.010
ĠletĠmeĠrecallĠthatĠtheĠrootsĠofĠx3Ġ=Ġ1ĠareĠ1,ĠÏī,ĠandĠÏ
Rollout 66, Act: 2.951
ĠaltitudeĠisĠC1.ĠNow,ĠforĠnĠâī¥Ġ2,ĠCnĠisĠtheĠfootĠofĠtheĠaltitude
Rollout 81, Act: 2.922
ĠtheĠaltitudeĠdividesĠBCĠintoĠsegmentsĠofĠ3ĠandĠ17,ĠthenĠtheĠtotalĠlengthĠofĠBCĠis
Rollout 32, Act: 2.894
,ĠsinceĠtheĠsumĠofĠrootsĠofĠx3Ġ-Ġ1Ġ=Ġ0ĠisĠzeroĠ(sinceĠtheĠcoefficient
Rollout 53, Act: 2.888
.5,ĠPDĠ=Ġ2,ĠDEĠ=Ġ2.5.ĠNeedĠtoĠfindĠtheĠareaĠof
Rollout 5, Act: 2.873
ĠbeĠb,ĠthenĠtheĠlongerĠbaseĠisĠbĠ+Ġ100.ĠTheĠmidlineĠlengthĠisĠ(
Rollout 30, Act: 2.845
)(b_nĠ+Ġ1)ĠforĠnĠâī¥Ġ2,ĊĊandĊĊb_{n+1}
Rollout 52, Act: 2.839
Ġb,ĠtheĠsumĠisĠ(bĠ-ĠaĠ+Ġ1)*(aĠ+Ġb)/2.ĠSo,
Rollout 65, Act: 2.821
ĠaĠfactorĠk,ĠwhereĠ0Ġ<ĠkĠ<Ġ1.ĠSimilarly,ĠifĠinstead,ĠtheĠlargerĠpiece
Rollout 12, Act: 2.819
{dx}{\sqrt{(x^2Ġ-Ġ1)(1Ġ-Ġ\nu^2Ġx^
Rollout 97, Act: 2.809
ĠN2O5.ĠWait,ĠbutĠtheĠproblemĠdoesn'tĠspecifyĠtheĠinitialĠamountĠinĠmoles.ĠIt
Rollout 44, Act: 2.805
ĠThenĠirreversibleĠworkĠisĠ2*(20Ġ-1)=38ĠL·atm.ĠReversible
Rollout 32, Act: 2.771
Ġunity).ĠSoĠÏīĠsatisfiesĠÏī3Ġ=Ġ1ĠandĠÏīĠâīłĠ1.ĠAlso,
Top 16 Negative Activations
Rollout 98, Act: -1.316
Ġcoexistence.ĊĊ3.Ġ**G2V,ĠK1V,ĠM5V**ĠĠĊĠĠĠ-
Rollout 21, Act: -1.330
(z),ĠfloorĠappliedĠtoĠzĠinĠ(-1,0)ĠgivesĠ-1.ĠSoĠf(z)Ġwould
Rollout 98, Act: -1.333
Ġ0.9MsunĠstarĠwouldĠbeĠK-type).ĊĊ**ValidĠSystems**:Ġ2Ġ(WDĠ+
Rollout 98, Act: -1.336
0V),Ġ3Ġ(G2V,ĠK1V,ĠM5V),Ġ4Ġ(
Rollout 98, Act: -1.349
existence.ĊĊ3.Ġ**G2V,ĠK1V,ĠM5V**ĠĠĊĠĠĠ-Ġ**
Rollout 98, Act: -1.408
ĠonĠtheĠMSĠwouldĠbeĠaĠlateĠGĠorĠearlyĠKĠstar.ĠTheĠproblemĠhereĠisĠifĠit'sĠcalled
Rollout 98, Act: -1.412
Ġexist.ĊĊThirdĠsystem:ĠG2V,ĠK1V,ĠM5V.ĠAllĠmain-sequence
Rollout 21, Act: -1.427
Ġ-Ġ(n+1)âĪĪĠ(-1,0).ĠHenceĠzĠinĠ(-1,0),ĠFloor
Rollout 21, Act: -1.443
1,0).ĠHenceĠzĠinĠ(-1,0),ĠFloor(z)ĠisĠ-1,ĠsoĠfloor
Rollout 98, Act: -1.478
.8ĠGyr).ĠAdditionally,ĠanĠ**A-typeĠstar**ĠcannotĠhaveĠ0.9Msun
Rollout 98, Act: -1.542
ĠandĠA0VĊĊG2V,ĠK1V,ĠM5VĊĊDA4,ĠL
Rollout 21, Act: -1.547
Ġ-Ġ(n+1)ĠâĪĪĠ(-1,0).ĠAndĠfloor(z),ĠfloorĠappliedĠtoĠzĠin
Rollout 21, Act: -1.558
Ġ-Ġ(k+1))Ġ=Ġ[-1,0).ĠApplyingĠfĠtoĠthat:Ġceil([-1,
Rollout 21, Act: -1.574
ceil(z).ĠButĠforĠzĠâĪĪĠ(-1,0),Ġceil(z)Ġ=0.ĠSoĠceil(x
Rollout 21, Act: -1.623
Ġ-Ġ(n+1)ĠâĪĪĠ(-1,0).Ġf(z)=ceil(z).ĠButĠforĠz
Rollout 57, Act: -1.713
1).ĠTherefore,ĠCĠ=Ġ(-infty,0]ĠâĪªĠ[1,infty).ĠNow,

UP_PROJ

Top 16 Positive Activations
Rollout 20, Act: 2.298
Ġ903.Ġ903ĠdividedĠbyĠ3Ġ=Ġ301.Ġ3
Rollout 89, Act: 2.078
0/3.ĠButĠ500ĠdividedĠbyĠ3ĠisĠaboutĠ166.6
Rollout 20, Act: 2.048
Ġ=Ġ903.Ġ903ĠdividedĠbyĠ3Ġ=Ġ301.Ġ
Rollout 70, Act: 1.990
147=6.ĠThenĠ6ĠdividedĠbyĠ7=0,ĠsoĠdigitsĠshouldĠbeĠ
Rollout 48, Act: 1.926
25ĠremĠ9.Ġ225/16=14ĠremĠ1.Ġ
Rollout 90, Act: 1.912
90.ĠIsĠ1090ĠdivisibleĠbyĠ2018?ĠLet'sĠcheck:
Rollout 90, Act: 1.905
426.Ġ1426ĠdividedĠbyĠ2018:Ġ201
Rollout 39, Act: 1.902
ĠbyĠ9.ĠSinceĠ999ĠdividedĠbyĠ9ĠisĠ111,Ġ9
Rollout 89, Act: 1.883
.Ġgirls=66Ġ(2/5ĠofĠ165=66),ĠbutĠboys=
Rollout 37, Act: 1.872
Ġ450.Ġ450ĠdividedĠbyĠ2!Ġ(whichĠisĠ2)Ġis
Rollout 32, Act: 1.871
NumberĠofĠcycles:Ġ2009ĠdividedĠbyĠ3ĠgivesĠ669ĠfullĠcyclesĠ(
Rollout 48, Act: 1.866
5.Ġ34,125ĠdividedĠbyĠ16ĠisĠ2132*
Rollout 92, Act: 1.865
517.Ġ1517ĠdividedĠbyĠ17200?Ġ17
Rollout 89, Act: 1.858
6Ġ(2/5ĠofĠ165=66),ĠbutĠboys=99.ĠNot
Rollout 0, Act: 1.851
Ġ2.ĊĊWait,Ġ7^2ĠdividesĠintoĠ20!ĠbutĠ7^3Ġdoes
Rollout 89, Act: 1.832
2/5.ĠSoĠ2/5ĠofĠ100Ġgirls.ĠButĠifĠgirlsĠareĠdefined
Top 16 Negative Activations
Rollout 27, Act: -2.532
Alternatively,ĠsupposeĠthatĠatĠT_c,ĠtheĠthermalĠenergyĠkĠT_cĠisĠequalĠtoĠtheĠquantumĠenergyĠper
Rollout 27, Act: -2.540
ms?ĊĊAlternatively,ĠmaybeĠtheĠproblemĠassumesĠthatĠtheĠthermalĠenergyĠatĠT_cĠisĠrelatedĠtoĠtheĠkineticĠenergy
Rollout 22, Act: -2.547
.ĊĊWait,ĠIĠcanĠreferĠtoĠtheĠformulaĠviaĠthermalĠdeĠBroglieĠwavelength;ĠforĠhydrogen.ĊĊAlternatively
Rollout 37, Act: -2.553
ĠisĠconnectedĠtoĠtheĠNewtonâĢĻsĠidentities,ĠrelatingĠelementaryĠsymmetricĠfunctionsĠtoĠpowerĠsums.ĊĊWait,ĠanotherĠapproach:
Rollout 27, Act: -2.554
ĠThen,ĠusingĠtheĠtotalĠrmsĠspeedĠtoĠcalculateĠtheĠthermalĠtemperatureĠviaĠ(1/2)mĠV_Be
Rollout 27, Act: -2.574
ĠproblemĠmightĠhaveĠtypoedĠandĠswappedĠrmsĠenergyĠwithĠthermalĠenergyĠperĠparticleĠatĠT_c.ĠSoĠÎĶE
Rollout 27, Act: -2.577
)kT,ĠwhichĠisĠhalfĠtheĠaverageĠthermalĠenergyĠperĠdegreeĠofĠfreedom.ĊĊWait,ĠtheĠproblemĠsays
Rollout 27, Act: -2.590
2)kT,ĠwhichĠisĠhalfĠtheĠaverageĠthermalĠenergyĠperĠdegreeĠofĠfreedom.ĊĊWait,ĠtheĠproblem
Rollout 4, Act: -2.591
ĠareĠindependentĠandĠrandom,ĠandĠcombinesĠthemĠinĠquadrature,ĠwhileĠtheĠmax-minĠapproachĠassumesĠthatĠallĠerrors
Rollout 37, Act: -2.593
ĠanyĠsymmetricĠpolynomialĠcanĠbeĠexpressedĠinĠtermsĠofĠelementaryĠsymmetricĠpolynomials.ĠTheĠdeterminantĠisĠanĠalternatingĠpolynomial,
Rollout 27, Act: -2.622
ĠWait,ĠwhatĠifĠtheĠproblemĠisĠmistakingĠtheĠthermalĠtemperatureĠcalculatedĠhereĠasĠT_c?ĠIfĠthat's
Rollout 27, Act: -2.633
ĠsimplifiedĠandĠassumesĠthatĠtheĠcriticalĠtemperatureĠisĠwhereĠtheĠthermalĠdeĠBroglieĠwavelengthĠisĠcomparableĠtoĠinterparticle
Rollout 27, Act: -2.664
?ĊĊAlternatively,ĠmaybeĠtheĠproblemĠassumesĠthatĠtheĠthermalĠenergyĠatĠT_cĠisĠrelatedĠtoĠtheĠkineticĠenergyĠvia
Rollout 27, Act: -2.677
?ĊĊAssumingĠthatĠthisĠV_BeĠisĠtheĠthermalĠvelocityĠatĠtemperatureĠT_c,ĠthenĠusingĠtheĠequip
Rollout 27, Act: -2.721
ĠofĠsightĠvelocityĠcausingĠaĠDopplerĠshiftĠinĠsomeĠenergyĠlevels?ĠWait,ĠbutĠtheĠproblemĠseemsĠtoĠwant
Rollout 27, Act: -2.724
ĠthenĠestimateĠT_c.ĠSoĠperhapsĠit'sĠaboutĠthermalĠmotionĠcausingĠsomeĠenergyĠfluctuations,ĠwhichĠrelatesĠtoĠtemperature

DOWN_PROJ

Top 16 Positive Activations
Rollout 66, Act: 2.769
ĠaltitudeĠisĠC1.ĠNow,ĠforĠnĠâī¥Ġ2,ĠCnĠisĠtheĠfootĠofĠtheĠaltitude
Rollout 16, Act: 2.567
ĠĊĊFirst,ĠthisĠproblemĠremindsĠmeĠofĠtheĠcouponĠcollectorĠproblem,ĠwhereĠyouĠhaveĠdifferentĠcouponsĠandĠyouĠwant
Rollout 54, Act: 2.412
CH2CH2COOâĪĴ)ĠisĠaĠresonance-stabilizedĠanion;ĠtheĠchargeĠisĠdel
Rollout 54, Act: 2.389
ĠaĠweakerĠbaseĠandĠlessĠnucleophilicĠbecauseĠofĠresonanceĠstabilization?ĊĊWait,ĠpropionateĠ(CH2CH
Rollout 16, Act: 2.260
allyĠdistributedĠvariablesĠisn'tĠstraightforward.ĠInĠtheĠcouponĠcollectorĠproblem,ĠeachĠnewĠcouponĠhasĠaĠdecreasingĠprobabilityĠof
Rollout 16, Act: 2.258
ĠexpectationĠwithĠindicators.ĊĊRecallĠthatĠforĠtheĠcouponĠcollectorĠproblemĠwithĠnĠcoupons,ĠtheĠexpectedĠtimeĠisĠn
Rollout 39, Act: 2.223
21}Ġ-Ġ1)/(10Ġ-1Ġ)Ġ=Ġ(10^{322}
Rollout 80, Act: 2.181
Ġ1ĠkmĠ=Ġ1000Ġ*Ġ100ĠcmĠ=Ġ10^5
Rollout 90, Act: 2.174
ĠbeĠupĠtoĠ10^{k}Ġ-1.ĠButĠNĠmustĠalsoĠbeĠsuchĠthatĠwhenĠmultiplied
Rollout 77, Act: 2.167
4)=[14Ġ/Ġ(14Ġ-10)]Ġ*Ġf(10,4)=
Rollout 77, Act: 2.165
AndĠIĠneedĠtoĠcomputeĠf(14,Ġ52).ĠHmm.ĠLet'sĠsee.ĊĊStarting
Rollout 16, Act: 2.119
Ġeach.ĠTheĠexpectedĠtimeĠshouldĠbeĠsimilarĠtoĠcouponĠcollector.ĠForĠcouponĠcollectorĠwithĠnĠcoupons,ĠtheĠexpectation
Rollout 32, Act: 2.119
ĠletĠmeĠrecallĠthatĠtheĠrootsĠofĠx3Ġ=Ġ1ĠareĠ1,ĠÏī,ĠandĠÏ
Rollout 2, Act: 2.101
)...(10^{999}Ġ-1).ĠWeĠneedĠtoĠcomputeĠPĠmodĠ10
Rollout 62, Act: 2.092
1)ĠgivesĠ11*(11Ġ-1Ġ+1)=11*11,Ġwhich
Rollout 54, Act: 2.088
'sĠmoreĠsolvated.ĠButĠpropionateĠisĠresonance-stabilizedĠâĨĴĠlessĠnucleophilic.ĠSo
Top 16 Negative Activations
Rollout 93, Act: -1.707
,ĠmaybeĠweĠcanĠapproximateĠeachĠtermĠusingĠLaplace'sĠmethodĠorĠsomethingĠsimilarĠforĠsums.ĠForĠlargeĠex
Rollout 99, Act: -1.711
.ĊĊWait,ĠperhapsĠanotherĠresourceĠsays:ĊĊInĠtheĠMüller'sĠmethod,ĠtoĠavoidĠinaccuracyĠfromĠsubtractionĠof
Rollout 12, Act: -1.723
Wait,Ġlet'sĠrecallĠthatĠwhenĠweĠuseĠdifferentiationĠunderĠtheĠintegralĠsign,ĠasĠlongĠasĠtheĠintegralĠconver
Rollout 69, Act: -1.751
.16e-5.ĠCos(30/27âīĪ1.111°
Rollout 51, Act: -1.752
ĠSoĠtheĠabsoluteĠvalueĠcanĠbeĠsplitĠasĠfollows:ĊĊ|sin(xĠ-Ġθ)|Ġ=Ġ{Ċsin(x
Rollout 99, Act: -1.755
ĠmethodĠworks.ĊĊFromĠwhatĠIĠremember,ĠinĠtheĠMüllerĠmethod,ĠtheĠnextĠestimateĠx3ĠisĠfoundĠby
Rollout 93, Act: -1.775
ĠequalsĠâĪijĠ[(expandĠintoĠbinomial,ĠexchangeĠorders,ĠthenĠsomeĠtermsĠsurvive.Ġ)ĊĊButĠtheĠsum
Rollout 99, Act: -1.781
Ġstep.ĠSo,Ġwe'reĠsupposedĠtoĠuseĠtheĠMüllerĠmethodĠtoĠperformĠtwoĠiterationsĠonĠtheĠequationĠlog1
Rollout 53, Act: -1.781
5ĊĊSo,ĠnowĠweĠhaveĠthreeĠequations:ĊĊ1.Ġ(aĠ+Ġc)^2Ġ+Ġd^
Rollout 11, Act: -1.792
Ġequations.ĠButĠwithoutĠrealĠroots.ĠButĠfromĠDescarteâĢĻsĠsigns,ĠifĠtheĠpolynomialĠwithĠrealĠcoefficientsĠhas
Rollout 16, Act: -1.792
/2)^{mk}ĊĊThen,ĠswappingĠtheĠorderĠofĠsummation,ĊĊE[T]Ġ=Ġsum_{
Rollout 16, Act: -1.836
2)^{mk}ĊĊThen,ĠswappingĠtheĠorderĠofĠsummation,ĊĊE[T]Ġ=Ġsum_{k
Rollout 99, Act: -1.839
,Ġokay.ĠĊĊFirst,ĠIĠrememberĠthatĠtheĠMüllerĠmethodĠisĠaĠnumericalĠmethodĠusedĠtoĠfindĠtheĠroots
Rollout 93, Act: -1.931
ĠmaybeĠweĠcanĠapproximateĠeachĠtermĠusingĠLaplace'sĠmethodĠorĠsomethingĠsimilarĠforĠsums.ĠForĠlargeĠexponents
Rollout 67, Act: -1.974
Ġ...(3)ĊĊSoĠnowĠweĠhaveĠthreeĠequations:ĊĊ1)ĠAĠ+ĠBĠ+ĠCĠ=Ġ14
Rollout 87, Act: -2.016
Ġx,ĠthenĠy,ĠthenĠz.ĠChangingĠtheĠorderĠshouldn'tĠmatterĠbecauseĠofĠtheĠsymmetry.ĊĊButĠintegrating

Layer 19

GATE_PROJ

Top 16 Positive Activations
Rollout 80, Act: 1.608
text{Volume}Ġ=Ġ\frac{4}{3}Ġ\piĠr^3Ġ=Ġ\frac{
Rollout 80, Act: 1.565
0^8Ġcm.ĊĊVolumeĠ=Ġ(4/3)ÏĢr3ĠâīĪĠ4/3
Rollout 80, Act: 1.498
ĠtheĠvolumeĠofĠtheĠinnerĠcoreĠisĠ(4/3)ÏĢr3.ĠLet'sĠcalculateĠthat:ĊĊ
Rollout 25, Act: 1.334
1Ġkm,ĠtheĠvolumeĠwouldĠbeĠ(4/3)ÏĢ*(3,541,0
Rollout 80, Act: 1.193
4Ġcm3.ĠThenĠmultiplyĠbyĠ(4/3)ÏĢ:ĠĊĊVolumeĠâīĪĠ(4/
Rollout 80, Act: 1.185
)ÏĢ:ĠĊĊVolumeĠâīĪĠ(4/3)Ġ*Ġ3.1416Ġ*
Rollout 32, Act: 1.184
$ĠbeĠaĠnonrealĠrootĠofĠ$x^3Ġ=Ġ1,$ĠandĠletĊ\[\math
Rollout 80, Act: 1.171
piĠr^3Ġ=Ġ\frac{4}{3}Ġ\piĠ(1.2Ġ\timesĠ
Rollout 51, Act: 1.152
ĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$\theta$ĠĠbeĠaĠconstantĠnumberĠsuchĠthatĠ
Rollout 80, Act: 1.146
3)ÏĢr3ĠâīĪĠ4/3Ġ*ĠÏĢĠ*Ġ(1.2ĠÃĹĠ1
Rollout 56, Act: 1.125
)Ġpasture.ĠSheĠstartsĠwithĠaĠtriangleĠ$A_{0}ĠA_{1}ĠA_{2}$
Rollout 80, Act: 1.124
24Ġcm3.ĊĊMultiplyĠbyĠ4/3ĠÏĢ:Ġ1.728Ġ*Ġ
Rollout 22, Act: 1.116
02214076e23Ġmolecules/mol):ĊĊ0.00201
Rollout 80, Act: 1.093
ĠLet'sĠcalculateĠthat:ĊĊVolumeĠ=Ġ(4/3)ÏĢ(1.2ĠÃĹĠ10
Rollout 36, Act: 1.066
ĊĊDivideĠallĠtermsĠbyĠ3:ĊĊs4Ġ-Ġ30,624s2
Rollout 11, Act: 1.061
,b,$ĠĠandĠĠ$c$ĠĠareĠthreeĠcomplexĠnumbersĠwithĠproductĠĠ$1$Ġ.
Top 16 Negative Activations
Rollout 40, Act: -3.078
5ÏĢ/2ĠdividedĠbyĠ2ÏĢĠisĠ75/4=18.75
Rollout 75, Act: -3.083
Wait,Ġ32ĠdividedĠbyĠ4ĠisĠ8.ĠSoĠ8ĠcanĠbeĠwrittenĠasĠsum
Rollout 20, Act: -3.085
021.ĠWait,ĠbutĠ1ĠmultipliedĠbyĠ1ĠisĠ1,ĠsoĠthatĠwouldĠwork
Rollout 75, Act: -3.101
1Ġ+Ġ3*0Ġ=Ġ3ĠmodĠ4.ĠAlternatively,ĠifĠweĠhaveĠsevenĠsquaresĠ(
Rollout 50, Act: -3.107
kĊĊWait,Ġ20/2ĠisĠ10.ĠSo:ĊĊ11Ġ=Ġ1
Rollout 69, Act: -3.122
ĠwhenĠi=0:Ġ30°Ġ/Ġ3^0Ġ=Ġ30°,Ġright
Rollout 84, Act: -3.132
3/16ĠequalsĠ2/9ĠofĠ27/32.ĠSoĠallĠchecksĠout
Rollout 96, Act: -3.137
1740ĠdividedĠbyĠ13ĠisĠ1740Ġ/13ĠisĠapproximatelyĠ
Rollout 93, Act: -3.153
Ġ2014ĠdividedĠbyĠ2ĠisĠ1007,ĠandĠ100
Rollout 71, Act: -3.175
Ġ2011Ġ÷Ġ3ĠisĠ670.333...ĠsoĠnot
Rollout 89, Act: -3.198
2/5.ĠSoĠ2/5ĠofĠ100Ġgirls.ĠButĠifĠgirlsĠareĠdefined
Rollout 89, Act: -3.218
.Ġgirls=66Ġ(2/5ĠofĠ165=66),ĠbutĠboys=
Rollout 89, Act: -3.229
ĠLet'sĠsee,Ġgirls=2/5ĠofĠ150=60.Ġboys=9
Rollout 71, Act: -3.303
Ġ8044ĠdividedĠbyĠ4ĠisĠ2011.ĠSoĠ3c2
Rollout 69, Act: -3.341
ĠwhenĠi=0:Ġ10°Ġ/Ġ1Ġ=Ġ10°,ĠandĠ3
Rollout 40, Act: -3.380
,Ġ150ĠdividedĠbyĠ4ĠisĠ37.5,ĠsoĠ75ÏĢ

UP_PROJ

Top 16 Positive Activations
Rollout 87, Act: 1.952
/2ĠandĠrĠfromĠ0ĠtoĠsqrt(2).ĊĊTheĠproblemĠthenĠisĠsplittingĠtheĠintegralĠintoĠtwo
Rollout 12, Act: 1.835
Ġt2)Ġ)Ġ~Ġ1Ġ/Ġsqrt(2ν(tĠ-Ġν)(1Ġ-Ġν2)
Rollout 12, Act: 1.735
ĠintegrandĠbecomesĠ~Ġ1Ġ/Ġsqrt(Ġ2νĠuĠ(1Ġ-Ġν2)Ġ),Ġso
Rollout 12, Act: 1.705
ĠtheĠlowerĠlimitĠisĠν,ĠupperĠlimitĠis1,ĠandĠintegrandĠdependsĠonĠν.ĊĊSoĠbyĠLe
Rollout 87, Act: 1.703
ĠtheĠfirstĠquadrantĠupĠtoĠaĠradiusĠofĠsqrt(2)*a).ĠWait,ĠbutĠintegratingĠfromĠy=
Rollout 87, Act: 1.629
Ġsquare,ĠrĠrangesĠfromĠ0ĠtoĠsqrt(2),ĠandĠθĠfromĠ0ĠtoĠÏĢ/
Rollout 9, Act: 1.608
,Ġy).ĠButĠtheĠtriangleĠbeingĠisoscelesĠwithĠbaseĠEM.ĠSinceĠtriangleĠGEMĠisĠis
Rollout 5, Act: 1.605
(18125)Ġ=Ġsqrt(25Ġ*Ġ725)Ġ=Ġ5
Rollout 54, Act: 1.601
9-20)ĠâĨĴĠweakerĠacidĠâĨĴĠstrongerĠbase.ĊĊ2.ĠHydroxideĠ(pKaĠ~
Rollout 92, Act: 1.593
ĠfromĠeye),ĠandĠtheĠimageĠisĠatĠtheĠnearĠpointĠ(63ĠcmĠfromĠeye).ĠHowever,Ġneed
Rollout 95, Act: 1.588
/3675^2)*sqrt(3675^2Ġ+671*(
Rollout 87, Act: 1.581
ĠintegralĠequalsĠ3Ġln(1Ġ+Ġsqrt(2))Ġ+Ġsqrt(3)Ġ-Ġ2.
Rollout 81, Act: 1.567
9Ġ+Ġh2)ĠandĠAC=sqrt(289Ġ+Ġh2).ĠThenĠangleĠatĠA
Rollout 54, Act: 1.559
ĠpolarĠproticĠsolventĠfollowsĠbasicityĠdueĠtoĠchargeĠdensityĠdominating.ĠSoĠmoreĠbasicĠO-centeredĠanionsĠare
Rollout 81, Act: 1.553
0Ġ(BC),Ġb=AC=Ġsqrt(289Ġ+h2),Ġc=AB=
Rollout 92, Act: 1.550
negativeĠbecauseĠit'sĠaĠvirtualĠimageĠonĠtheĠsameĠsideĠasĠtheĠobject).ĠĊĊSo,ĠpluggingĠinto
Top 16 Negative Activations
Rollout 51, Act: -2.013
}ĊĊButĠsinceĠourĠintegralĠisĠoverĠ0ĠtoĠ2ÏĢ,ĠwhichĠcanĠbeĠpartitionedĠintoĠ[
Rollout 5, Act: -2.015
ĠsegmentĠconnectingĠtheĠmidpointsĠofĠtheĠlegsĠdividesĠtheĠtrapezoidĠintoĠtwoĠregionsĠwithĠareasĠinĠthe
Rollout 55, Act: -2.015
ĠhaveĠthisĠmatrixĠAĠandĠIĠneedĠtoĠcomputeĠAĠraisedĠtoĠtheĠ2018thĠpower.
Rollout 56, Act: -2.017
ĠstartsĠwithĠaĠtriangleĠ$A_{0}ĠA_{1}ĠA_{2}$ĠwhereĠangleĠ$A
Rollout 49, Act: -2.017
ĠstatementĠis:ĠIfĠeveryĠsubspaceĠofĠaĠHausdorffĠspaceĠXĠisĠÏĥ-compact,Ġthen
Rollout 20, Act: -2.023
licativeĠgroupĠmoduloĠm.ĠWait,ĠthatĠis,ĠknĠmustĠbeĠaĠmultipleĠofĠtheĠexponentĠofĠtheĠmultip
Rollout 40, Act: -2.025
Ġyi,ĠthenĠaĠrotationĠbyĠÏĢ/4ĠcounterclockwiseĠaroundĠtheĠoriginĠwouldĠbeĠmultiplyingĠbyĠe
Rollout 20, Act: -2.043
ĠofĠeachĠprimeĠpower.ĊĊTheĠproblemĠwantsĠÏĥ(an)Ġ-Ġ1Ġâī¡Ġ0ĠmodĠ2
Rollout 46, Act: -2.065
:ĊĊTheĠcrossĠproductĠtermĠÏīĠÃĹĠv,ĠinĠcomponentsĠassumingĠÏīĠisĠalongĠtheĠz-axis:
Rollout 20, Act: -2.071
)^ÃĹĠisĠcyclicĠofĠorder42,ĠneedĠknĠâī¡0Ġmod42.ĠButĠforĠkn
Rollout 10, Act: -2.082
ĊĊWait,Ġokay,ĠsoĠfromĠ0ĠtoĠ10,ĠtheĠlosingĠpositionsĠ(P-positions
Rollout 49, Act: -2.090
,ĠaĠcommonĠexampleĠofĠanĠuncountableĠHausdorffĠspaceĠthatĠisn'tĠÏĥ-compactĠis
Rollout 2, Act: -2.095
ĠremainderĠwhenĠtheĠproductĠofĠnumbersĠlikeĠ9,Ġ99,Ġ999,ĠandĠso
Rollout 65, Act: -2.099
.ĠHmmm.ĊĊWait,ĠperhapsĠweĠneedĠthatĠafterĠcuttingĠaĠprismĠbyĠaĠplaneĠparallelĠtoĠoneĠof
Rollout 20, Act: -2.113
ButĠholdĠon,Ġm=43:ĠneedĠknĠâī¡Ġ0ĠmodĠ42,Ġand
Rollout 20, Act: -2.122
ĠthatĠexponentĠisĠ1.ĊĊTherefore,ĠweĠneedĠknĠâī¡Ġ0ĠmodĠmĠ-Ġ1,

DOWN_PROJ

Top 16 Positive Activations
Rollout 49, Act: 3.958
ĠshowĠthatĠifĠXĠisĠanĠuncountableĠHausdorffĠspace,ĠthenĠthereĠexistsĠsomeĠsubspace
Rollout 32, Act: 3.939
ĠletĠmeĠrecallĠthatĠtheĠrootsĠofĠx3Ġ=Ġ1ĠareĠ1,ĠÏī,ĠandĠÏ
Rollout 68, Act: 3.880
ĠhasĠtoĠbeĠpositive.ĠSimilarly,Ġlog(xĠ+Ġ2)ĠisĠdefinedĠonlyĠwhenĠxĠ+Ġ2
Rollout 96, Act: 3.703
ĠperpendicularĠtoĠLĠandĠcontainsĠpointĠB(5,Ġ6).Ċ-ĠInĠtheĠnewĠcoordinateĠsystem,Ġline
Rollout 18, Act: 3.668
ĠaĠpairĠ(x,Ġy)ĠwithĠxĠ>Ġ0ĠsuchĠthatĠHĠisĠequalĠtoĠH_{xy
Rollout 11, Act: 3.652
1/cĠ=Ġab.ĠSoĠ1/aĠ+Ġ1/bĠ+Ġ1/cĠ=ĠbcĠ+Ġac
Rollout 32, Act: 3.648
Ġunity).ĠSoĠÏīĠsatisfiesĠÏī3Ġ=Ġ1ĠandĠÏīĠâīłĠ1.ĠAlso,
Rollout 49, Act: 3.499
,ĠaĠcommonĠexampleĠofĠanĠuncountableĠHausdorffĠspaceĠthatĠisn'tĠÏĥ-compactĠis
Rollout 47, Act: 3.448
ĠSt,ĠwhileĠtheĠtunnel'sĠisĠS0Ġ=Ġ4St.ĠThatĠmeansĠtheĠareaĠaroundĠtheĠtrain
Rollout 69, Act: 3.429
ĠtheĠsineĠandĠcosineĠareĠ10°ĠandĠ30°ĠdividedĠbyĠ3^i,
Rollout 11, Act: 3.425
Ġ+ĠbĠ+Ġc)Ġ+Ġ(1/aĠ+Ġ1/bĠ+Ġ1/c),ĠandĠqĠas
Rollout 33, Act: 3.394
,6}.ĠForĠeachĠkĠfromĠ1ĠtoĠ5,ĠwhenĠweĠlookĠatĠtheĠfirstĠkĠelements
Rollout 66, Act: 3.392
ĠaltitudeĠisĠC1.ĠNow,ĠforĠnĠâī¥Ġ2,ĠCnĠisĠtheĠfootĠofĠtheĠaltitude
Rollout 31, Act: 3.390
+1}Ġ+ĠyĠa_nĠforĠnĠâī¥Ġ0,ĠwhereĠxĠandĠyĠareĠrealĠnumbers.
Rollout 30, Act: 3.386
-1}Ġ-Ġ1)(b_nĠ+Ġ1).ĠThisĠrelatesĠa_{n+1}
Rollout 11, Act: 3.384
ĠnotĠsure.ĠAlternatively,ĠpĠisĠ(aĠ+Ġ1/a)Ġ+Ġ(bĠ+Ġ1/b
Top 16 Negative Activations
Rollout 99, Act: -2.037
Ġis2.1573.ĠHence,ĠboxedĠanswerĠasĠ\boxed{2.157
Rollout 89, Act: -2.038
ĠgirlsĠareĠinĠtheĠclass?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠseeĠhere.ĠSoĠthe
Rollout 52, Act: -2.040
ĠisĠtheĠsmallestĠyummyĠinteger?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠaboutĠyummy
Rollout 25, Act: -2.048
$ĠisĠtheĠimpactĠparameter.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠthisĠproblem
Rollout 80, Act: -2.071
)ÏĢ:ĠĊĊVolumeĠâīĪĠ(4/3)Ġ*Ġ3.1416Ġ*
Rollout 80, Act: -2.084
3)ÏĢr3ĠâīĪĠ4/3Ġ*ĠÏĢĠ*Ġ(1.2ĠÃĹĠ1
Rollout 80, Act: -2.105
piĠr^3Ġ=Ġ\frac{4}{3}Ġ\piĠ(1.2Ġ\timesĠ
Rollout 83, Act: -2.113
ĠtheĠsameĠifĠyouĠcanĠrotateĠoneĠtoĠgetĠtheĠother.ĠSoĠweĠhaveĠtoĠconsiderĠrotationalĠsymmetry.ĊĊSo
Rollout 25, Act: -2.210
00Ġm,ĠthenĠmĠ=Ġ(4/3)ÏĢ*(3.541e6
Rollout 98, Act: -2.221
Ċ(D)Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠhow
Rollout 80, Act: -2.377
ĠLet'sĠcalculateĠthat:ĊĊVolumeĠ=Ġ(4/3)ÏĢ(1.2ĠÃĹĠ10
Rollout 80, Act: -2.390
0^8Ġcm.ĊĊVolumeĠ=Ġ(4/3)ÏĢr3ĠâīĪĠ4/3
Rollout 25, Act: -2.407
ĠÏģ,ĠthenĠmassĠmĠ=Ġ(4/3)ÏĢR3Ïģ.ĠSoĠforĠRĠ=
Rollout 80, Act: -2.437
text{Volume}Ġ=Ġ\frac{4}{3}Ġ\piĠr^3Ġ=Ġ\frac{
Rollout 25, Act: -2.558
1Ġkm,ĠtheĠvolumeĠwouldĠbeĠ(4/3)ÏĢ*(3,541,0
Rollout 80, Act: -2.737
ĠtheĠvolumeĠofĠtheĠinnerĠcoreĠisĠ(4/3)ÏĢr3.ĠLet'sĠcalculateĠthat:ĊĊ

Layer 20

GATE_PROJ

Top 16 Positive Activations
Rollout 85, Act: 2.545
ĠallĠnumbersĠareĠnonnegativeĠintegersĠandĠ0ĠisĠalreadyĠatĠ(0,0),ĠcanĠweĠhaveĠ
Rollout 20, Act: 2.506
ĠwhenĠyouĠsetĠk=1,ĠitĠmustĠhold,ĠbutĠthenĠforĠhigherĠk,ĠitĠfollowsĠautomatically.
Rollout 83, Act: 2.491
ĠallĠarrangementsĠhaveĠtrivialĠstabilizerĠ(noĠrotationĠfixĠanyĠarrangementĠwithĠuniqueĠdigits,ĠsoĠeveryĠorbitĠindeedĠhas
Rollout 95, Act: 2.447
3)Ġk)/490ĠmustĠequalsĠrationalĠnumberĠmĠâĨĴsqrt(3)=490*m
Rollout 10, Act: 2.439
ĠnumberĠyouĠaddĠtoĠS,ĠyouĠmustĠexcludeĠonlyĠfinitelyĠmanyĠothersĠ(i.e.,ĠnĠ+
Rollout 35, Act: 2.391
4)=5.ĠSinceĠ5ĠandĠ1ĠisĠprime.ĠThenĠsolutionsĠasĠfactorsĠwouldĠhaveĠsameĠissues
Rollout 21, Act: 2.382
.ĠButĠsuchĠfĠmustĠsatisfyĠforĠeachĠintegerĠk,Ġf(k)Ġ=xĠwhereĠxĠisĠmappedĠunder
Rollout 37, Act: 2.324
ĠisĠdivisibleĠbyĠ(1-2)=Ġ-1,ĠhenceĠifĠm=1,ĠthisĠtermĠisĠ-
Rollout 70, Act: 2.316
ĠtoĠbalanceĠtheĠequation,ĠtheĠhigherĠdigitsĠmustĠprovideĠenoughĠpositiveĠvalueĠtoĠoffsetĠtheĠnegativesĠfromĠtheĠlowerĠdigits
Rollout 65, Act: 2.310
95,Ġc.ĠForĠsimilarity,ĠthisĠmustĠsatisfyĠthatĠa1,Ġ1995,
Rollout 64, Act: 2.307
Ġnegative,ĠtheĠtermĠB*2^nĠneedsĠtoĠcompensateĠitĠforĠallĠn.ĊĊE.g.,ĠsupposeĠA
Rollout 65, Act: 2.297
Ġc2=1995*aĠmustĠholdĠwithĠaĠbeingĠ<=1995ĠandĠcĠbeing
Rollout 64, Act: 2.285
Ġ-KĠy.ĠWhichĠforĠeachĠyĠisĠexact,ĠsinceĠbothĠtermsĠareĠintegers.ĠThus,ĠCĠ+
Rollout 82, Act: 2.285
Ġlog2(2)Ġ(whichĠisĠ1),ĠbutĠit'sĠmoreĠcomplicated.ĠAlternatively,ĠifĠwe
Rollout 21, Act: 2.275
)ĠtoĠbeĠZĠbyĠmakingĠforĠeveryĠrealĠy,Ġf(y)ĠmustĠbeĠinĠZ.ĠButĠhow
Rollout 8, Act: 2.264
ĠareĠtheĠonlyĠpossibilitiesĠforĠmixed,ĠandĠeachĠisĠalreadyĠaĠblockĠofĠ1ĠredĠandĠ1Ġgreen
Top 16 Negative Activations
Rollout 40, Act: -2.549
Ġ3ÏĢ/2}Ġ=Ġcos(3ÏĢ/2)Ġ+ĠiĠsin(3ÏĢ/
Rollout 69, Act: -2.550
ĠphiĊĊButĠnotĠsure.ĊĊAlternatively,ĠwriteĠsin2(phi/3)Ġ=Ġ(1Ġ-Ġcos(
Rollout 93, Act: -2.554
ÏĢ/1)]^{2}.ĠSoĠcos(ÏĢ)^2Ġ=Ġ(-1)^2Ġ=1.
Rollout 93, Act: -2.556
)Ġ=Ġ-sinx.ĠTherefore,Ġcos(ÏĢ/2Ġ+ĠÏĢ/(2*100
Rollout 3, Act: -2.561
.ĠLet'sĠcheckĠwithĠcalculatorĠsteps.ĊĊAlternatively,Ġsqrt(28,408).ĠLet's
Rollout 93, Act: -2.580
1007}.ĠAndĠweĠknowĠthatĠcos2θĠcanĠbeĠwrittenĠasĠ(1Ġ+Ġcos2
Rollout 69, Act: -2.583
Ġhandle.ĊĊSoĠnow,ĠperhapsĠweĠcanĠexpressĠsin2(theta)ĠinĠtermsĠthatĠinvolveĠcos(3Ġtheta
Rollout 17, Act: -2.600
e-11)^3Ġ)ĊĊTheĠ4ÏĢe-7ĠisĠμ0,ĠwhichĠisĠ
Rollout 69, Act: -2.612
ĠtrigonometricĠidentities.ĠMaybeĠweĠcanĠwriteĠsin2(theta)ĠusingĠaĠdoubleĠangleĠidentity,ĠorĠexpress
Rollout 93, Act: -2.615
07)ĠandĠcos(1006ÏĢ/1007)Ġ=Ġ-cos(
Rollout 93, Act: -2.642
=0.ĊĊk=3:Ġcos(3ÏĢ/4)=Ġ-âĪļ2/2.Ġsame
Rollout 93, Act: -2.651
ĠlargeĠevenĠnumber,ĠmaybeĠIĠcanĠexpressĠcos^{2014}(θ)ĠinĠtermsĠofĠmultiple
Rollout 93, Act: -2.660
}^{1007}Ġ[cos(ÏĢk/1007)]^{20
Rollout 93, Act: -2.698
1006ĠwouldĠpairĠtoĠcos(2ÏĢ/1007)ĠandĠcos(1
Rollout 55, Act: -2.727
ĠandĠIĠneedĠtoĠcomputeĠAĠraisedĠtoĠtheĠ2018thĠpower.ĠThatĠseemsĠlikeĠa
Rollout 93, Act: -2.738
2*1007).ĠSoĠcos(ÏĢ/2Ġ+Ġx)Ġ=Ġ-sinx.

UP_PROJ

Top 16 Positive Activations
Rollout 48, Act: 2.085
ĠtheĠheaderĠcontainsĠerror?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠIPv4ĠheaderĠin
Rollout 3, Act: 2.052
69Ġm/s.ĊĊ**FinalĠAnswer**Ċ\boxed{170}Ġm/sĊĊWaitĠno
Rollout 54, Act: 2.046
ĠBĠ=Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestionĠstepĠbyĠstep
Rollout 3, Act: 2.045
3,ĠsoĠifĠneedĠtoĠanswer,ĠinĠtheĠabsenceĠofĠtheĠtemperatureĠinĠtheĠproblem,ĠIĠshouldĠeither
Rollout 3, Act: 2.044
.5Ġm/s.ĠIfĠusingĠcalculatorĠaccurateĠvalue.ĊĊSoĠtheĠanswerĠisĠapproximatelyĠ168.
Rollout 43, Act: 2.038
\right\rangleĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthisĠproblem
Rollout 3, Act: 2.030
69}Ġm/s.ĊĊ**FinalĠAnswer**ĊTheĠmeanĠmolecularĠspeedĠofĠradonĠgasĠisĠ\boxed
Rollout 1, Act: 2.017
ĠproblemĠstatement.ĊĊ**[FinalĠAnswer]**Ċ\boxed{\text{SuchĠaĠpointĠ}ĠyĠ\text
Rollout 56, Act: 2.013
ĠTherefore,ĠweĠconclude.ĊĊ**FinalĠAnswer**Ċ\boxed{\sqrt{3}}ĊĊ<|im_start|>answerĊWes
Rollout 4, Act: 2.009
Ġ0.88%.ĊĊ**FinalĠAnswer**ĊTheĠpercentageĠerrorĠinĠtheĠcalculationĠofĠtheĠenergyĠstoredĠis
Rollout 64, Act: 2.006
ĠisĠtheĠdesiredĠanswer.ĊĊ**FinalĠAnswer**Ċ\boxed{36Ġ\left(2^{20
Rollout 99, Act: 2.005
ĠofĠx_3?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthisĠproblem
Rollout 29, Act: 2.002
ĠAnswer**Ċ\boxed{A}ĊĊ<|im_start|>answerĊGivenĠaĠgallonĠmilkĠjugĠwithĠaĠpluggedĠhandleĠdueĠto
Rollout 22, Act: 1.999
ĠButĠinĠseveralĠdecimalĠplaces.ĊĊĊ**FinalĠAnswer**ĊTheĠmeanĠmolecularĠspeedĠofĠhydrogenĠgasĠ(H2)
Rollout 25, Act: 1.989
text{Ġkg}}$ĊĊĊ**FinalĠAnswer**ĊTheĠasteroidĠwouldĠneedĠtoĠbeĠ$\boxed{4.
Rollout 27, Act: 1.986
0^{6}Ġ\text{ĠK}}$.ĊĊ<|im_start|>answerĊGivenĠÎĶE_rmsĠ=Ġ
Top 16 Negative Activations
Rollout 16, Act: -1.978
ĠconvertedĠbyĠdayĠtĠisĠ(1/2)^t.ĠTherefore,ĠtheĠprobabilityĠthatĠallĠsixĠsaplings
Rollout 2, Act: -1.983
ĠThus,ĠforĠkĠâī¥3,Ġ2^kĠâī¡Ġ0Ġmod8,ĠsoĠ(2
Rollout 51, Act: -1.997
xĠ+Ġx)Ġ+Ġsin(xĠ-Ġ8x)]Ġ/Ġ2Ġ=Ġ[sin9xĠ+
Rollout 72, Act: -2.028
ĠrootĠsinceĠnĠisĠevenĠ(sinceĠ(-1)^n=1ĠwhenĠnĠeven).ĠTherefore,Ġyes,
Rollout 2, Act: -2.029
=3.ĠForĠk>=3:2^kĠmod8.ĠButĠ2^3=8
Rollout 50, Act: -2.049
20x)^kĠ=Ġ20^kĠx^kĠ\).ĠSoĠtheĠequationĠis:ĊĊ
Rollout 73, Act: -2.064
ĠsumĠ2^jĠ-ĠsumĠ2^iĠoverĠallĠjĠ>i.ĊĊButĠsumĠoverĠall
Rollout 45, Act: -2.065
1Ġ+Ġ(4Ġ-m)Ġ-Ġ(5mĠ+1)Ġ-4(mĠ+1)Ġâī¥
Rollout 93, Act: -2.075
m)!)/(m!^2Ġ*4^m}.ĊĊWeĠalsoĠknowĠthatĠaccordingĠtoĠcentralĠbin
Rollout 45, Act: -2.075
Ġ-4)Ġ+Ġ(-4mĠ-10mĠ-4m)=18Ġ-18m
Rollout 45, Act: -2.082
8Ġ+16Ġ-4mĠ-10mĠ-2Ġ-4mĠ-4=Ġ(8
Rollout 50, Act: -2.087
101Ġ*Ġ(20/2)^kĊĊWait,Ġ20/2ĠisĠ
Rollout 2, Act: -2.121
Ġ\geqĠ3\),Ġ\(2^kĠ\equivĠ0Ġ\modĠ8\),
Rollout 61, Act: -2.227
9.ĠLetâĢĻsĠwriteĠthisĠasĠ2^m(1Ġ+Ġ2^{kĠ-Ġm})
Rollout 45, Act: -2.323
Ġ-1Ġ-4)Ġ+Ġ(-mĠ-5mĠ-4m)=0Ġ-10mĠâī¥
Rollout 45, Act: -2.446
ĊĊCalculate:ĊĊ1Ġ+4Ġ-mĠ-5mĠ-1Ġ-4mĠ-4=Ġ(1

DOWN_PROJ

Top 16 Positive Activations
Rollout 62, Act: 2.212
-generatingĠpolynomial?ĠMaybeĠsimilarĠtoĠEuler'sĠfamousĠpolynomialĠn^2Ġ+nĠ+41,Ġwhich
Rollout 34, Act: 2.142
ĠoutĠagain:ĊĊ1.Ġ\(Ġx(t)Ġ=Ġ1Ġ+Ġ\int_{0}^{t}
Rollout 62, Act: 2.017
'sĠfamousĠpolynomialĠn^2Ġ+nĠ+41,ĠwhichĠgeneratesĠprimesĠforĠn=0ĠtoĠ
Rollout 54, Act: 1.978
ĠdifferentĠanions:ĠbulkierĠonesĠmightĠbeĠlessĠnucleophilicĠbecauseĠofĠstericĠhindrance.ĠAlso
Rollout 54, Act: 1.933
ĠfromĠtheĠfollowingĠstatementsĠ(B).ĊĊ1.ĠInĠsubstitutionĠreactionĠifĠtheĠreactantĠinĠrateĠdeterminingĠstepĠis
Rollout 54, Act: 1.932
ĠmoreĠstabilized,ĠbutĠthat'sĠconfusing)ĠareĠbetterĠnucleophilesĠwhenĠconsideringĠpolarizability.ĠOrĠis
Rollout 54, Act: 1.913
Ġactually,ĠstericĠhindranceĠmightĠmakeĠitĠlessĠnucleophilicĠbecauseĠtheĠloneĠpairsĠareĠharderĠtoĠaccess
Rollout 83, Act: 1.912
,Ġhere'sĠaĠproperĠwayĠtoĠcalculate:ĊĊBurnsideâĢĻsĠformula:ĠcountĠofĠorbits=Ġ(SumĠover
Rollout 50, Act: 1.898
.ĠAlso,Ġ22ĠisĠ2ĠÃĹĠ11.ĠMaybeĠtheseĠfactorizationsĠcanĠhelpĠin
Rollout 54, Act: 1.897
ĠaproticĠsolvents,ĠalkoxideĠisĠmoreĠnucleophilic,ĠbutĠinĠpolarĠprotic,Ġdue
Rollout 54, Act: 1.886
ĠandĠattack.ĠAlso,ĠweakerĠbasesĠmightĠbeĠpoorerĠnucleophilesĠinĠpolarĠproticĠsolventsĠbecauseĠstronger
Rollout 54, Act: 1.885
Ġbasicity.ĠStericĠhindranceĠalsoĠreducesĠnucleophilicity.ĠTheĠorderĠis:ĠĠĊĊ**Ethan
Rollout 54, Act: 1.880
ĠionsĠ(e.g.,Ġsulfur-based)ĠareĠbetterĠnucleophilesĠdespiteĠweakerĠbasicity.ĠStericĠhind
Rollout 54, Act: 1.877
ĠbasesĠareĠmoreĠsolvated,ĠsoĠtheirĠnucleophilicityĠisĠreduced.ĠHowever,ĠpolarizabilityĠalso
Rollout 69, Act: 1.873
ĠtheĠsineĠandĠcosineĠareĠ10°ĠandĠ30°ĠdividedĠbyĠ3^i,
Rollout 76, Act: 1.872
ĠoneĠmightĠthinkĠtheĠtotalĠnumberĠisĠ4Ġ*Ġ32Ġ=Ġ128.ĠButĠwait
Top 16 Negative Activations
Rollout 30, Act: -1.666
_{n+1Ġ}ĊĊLetĠmeĠcheckĠifĠthisĠcanĠtelescope.ĠFromĠtheĠrecurrenceĠrelations.ĠLet'sĠrecall
Rollout 27, Act: -1.684
ĠvalueĠofĠtheĠcomponentĠofĠvelocityĠalongĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġso
Rollout 29, Act: -1.696
'sĠbottom.ĠThus,ĠtheĠanswerĠwouldĠbeĠA)ĠPĠ=ĠrhoĠgĠh.ĠButĠaccordingĠtoĠoptions
Rollout 3, Act: -1.697
ĠofĠsomeĠstandardĠconditions.ĠHmm.ĊĊWait,ĠinĠkineticĠtheory,ĠmeanĠspeedĠdependsĠonĠtemperature,ĠsoĠwe
Rollout 70, Act: -1.701
7=3,Ġremainder0;ĠthenĠ3ĠdividedĠby7=0,Ġremainder3.ĠSoĠdigits
Rollout 72, Act: -1.704
onometricĠidentities,ĠweĠknowĠthatĠcos(72°)Ġ=Ġ2Ġcos^2(3
Rollout 27, Act: -1.720
ĠsightĠisĠoneĠdegreeĠofĠfreedom,ĠthenĠtheĠaverageĠkineticĠenergyĠforĠthatĠcomponentĠisĠ(1/2)
Rollout 69, Act: -1.728
(3Ġtheta_i).ĠLet'sĠseeĠifĠthisĠcanĠbeĠmanipulated.ĊĊPerhapsĠuseĠtheĠidentityĠforĠcos(
Rollout 27, Act: -1.728
?ĊĊAlternatively,ĠmaybeĠit'sĠtheĠlineĠofĠsightĠvelocityĠcausingĠaĠDopplerĠshiftĠinĠsomeĠenergyĠlevels?
Rollout 47, Act: -1.745
Ġ$v_{t}$ĠinsideĠaĠlongĠtunnelĠofĠcross-sectionalĠareaĠ$S_{0}\left(S_{
Rollout 27, Act: -1.758
.ĠAlternatively,ĠifĠtheĠmotionĠalongĠtheĠlineĠofĠsightĠcausesĠaĠDopplerĠshiftĠinĠsomeĠemittedĠorĠabsorbed
Rollout 27, Act: -1.761
ĠtoĠvelocity.ĠIfĠtheĠvelocityĠalongĠtheĠlineĠofĠsightĠisĠv_z,ĠthenĠtheĠDopplerĠshiftĠcould
Rollout 89, Act: -1.794
Ġnumber,ĠthenĠtakeĠ2/5ĠofĠthatĠtoĠgetĠtheĠnumberĠofĠgirls.ĠLet'sĠconfirmĠthat
Rollout 27, Act: -1.822
ĠtheĠthermalĠenergyĠatĠT_cĠisĠrelatedĠtoĠtheĠkineticĠenergyĠviaĠ(3/2)kT_c
Rollout 95, Act: -1.860
ĠtheĠheightĠofĠtheĠtriangleĠABCĠwithĠrespectĠtoĠbaseĠBC.ĠSinceĠareaĠisĠ3âĪļ671
Rollout 27, Act: -1.896
1/2)ĠkTĠasĠlineĠofĠsightĠkineticĠenergy,ĠthenĠTĠ=Ġ2*ÎĶE

Layer 21

GATE_PROJ

Top 16 Positive Activations
Rollout 54, Act: 2.800
Ġsolution:ĊĊInĠSN2Ġreactions,ĠtheĠthiolateĠionĠ(e.g.,ĠSHâĪĴ)ĠisĠa
Rollout 54, Act: 2.767
ĠstrongerĠacidsĠthanĠalcohols,ĠsoĠthiolateĠisĠaĠweakerĠbase?ĠWait,ĠpKaĠof
Rollout 54, Act: 2.701
Ġ(CH3OH)Ċ5.ĠEthanethiolateĠ(HSâĪĴĠorĠmaybeĠC2H5S
Rollout 54, Act: 2.586
5SâĪĴ?ĠWait,ĠethanethiolateĠwouldĠbeĠCH3CH2SâĪĴ)ĊĊWait
Rollout 54, Act: 2.580
.ĠMethanol,Ġ5.ĠEthanethiolate)ĠfromĠmostĠtoĠpoorestĠreactivityĠinĠaqueous
Rollout 54, Act: 2.576
?Ġ~10,ĠsoĠethanethiolateĠ(conjugateĠbase)ĠisĠweakerĠbaseĠthan
Rollout 54, Act: 2.537
ĠevenĠinĠpolarĠproticĠsolvents,ĠthiolatesĠ(likeĠSHâĪĴ)ĠareĠbetterĠnucleophiles
Rollout 54, Act: 2.519
ĠinĠaproticĠsolvent,Ġtert-butoxideĠwouldĠbeĠbetter.ĊĊSo,ĠinĠpolarĠprotic
Rollout 54, Act: 2.514
Ġ(strongĠbase)Ċ3.Ġethanethiolate?ĠWait,Ġwhat'sĠtheĠpKaĠofĠeth
Rollout 54, Act: 2.514
3CH2SH,ĠsoĠethanethiolateĠwouldĠbeĠCH3CH2SâĪĴ.ĠOkay
Rollout 54, Act: 2.501
imolecularĠattack,ĠsoĠifĠtheĠnucleophileĠisĠcharged,ĠsolvationĠcanĠhinderĠitsĠreactivity.ĠBut
Rollout 54, Act: 2.487
ĠsameĠtypeĠofĠnucleophiles:Ġoxygen-basedĠanions.ĠBetweenĠhydroxideĠ(small,ĠlessĠpolar
Rollout 54, Act: 2.484
ionate.ĠThatĠmakesĠsense.ĊĊEthanethiolateĠ(HSâĪĴĠorĠRSâĪĴ):ĠSulfur
Rollout 49, Act: 2.476
?ĠWait,Ġno,ĠÏĥ-compactĠimpliesĠLindelöf,ĠbutĠLindelöfĠplusĠmet
Rollout 54, Act: 2.468
:ĊĊ1.ĠIfĠreactantĠinĠRDSĠisĠchargedĠ(probablyĠmeaningĠtheĠstartingĠmaterialĠisĠcharged),Ġand
Rollout 49, Act: 2.461
?ĊĊNo.ĠThereĠareĠHausdorffĠLindelöfĠspacesĠthatĠareĠnotĠregular,ĠcalledĠT
Top 16 Negative Activations
Rollout 64, Act: -3.097
2009}Ġ-1Ġ)ĊĊAlternatively,ĠtoĠsolveĠforĠa2,ĠletĠmeĠwriteĠtheĠinequality
Rollout 2, Act: -3.115
Ġof1000Ġhere?ĊĊAlternatively,ĠmaybeĠbecauseĠmanyĠofĠtheĠtermsĠinĠtheĠproductĠfromĠk=
Rollout 23, Act: -3.130
0).ĠNo,ĠmixĠofĠterms.ĊĊAlternatively,ĠthinkĠofĠActivityĠremainsĠasĠinitialĠofĠlambda_alphaĠ*N
Rollout 93, Act: -3.132
m}.ĊĊĠInĠourĠderivationĠprocess,mismatchĠaroseĠdueĠtoĠtheĠformulaĠS=Ġm*(2Ġ+ĠC
Rollout 65, Act: -3.153
Ġx,ĠbutĠthisĠseemsĠnotĠstraightforward.ĠAlternatively,ĠthinkĠinĠtermsĠofĠproportionality.ĠIfĠweĠhaveĠsides
Rollout 8, Act: -3.199
ĠareĠadjacentĠandĠallĠgreensĠareĠadjacent.ĊĊAlternatively,ĠthinkĠofĠtheĠproblemĠasĠtheĠlaid-outĠ5Ġcards
Rollout 69, Act: -3.200
ĠandĠtheta_{i+1}.ĊĊAlternatively,ĠthinkĠaboutĠpossibleĠtelescopingĠproducts.ĠButĠtheĠsumĠis
Rollout 73, Act: -3.207
Ġ-k}.ĠWait,ĠmaybeĠnot.ĊĊAlternatively,ĠthinkĠofĠitĠasĠsum_{i=0}^{
Rollout 96, Act: -3.216
ĠtoĠmakeĠsureĠtheĠorientationĠisĠcorrect.ĠAlternatively,ĠmaybeĠtheĠangleĠforĠrotationĠshouldĠaccountĠforĠtheĠdirection.ĊĊ
Rollout 83, Act: -3.258
ĠmeĠcross-verifyĠwithĠanotherĠapproach.ĊĊAlternatively,ĠthinkĠofĠarrangingĠtheĠresiduesĠinĠaĠcycle.ĠSinceĠeach
Rollout 76, Act: -3.266
'sĠaĠmistakeĠinĠinclusion-exclusion.ĊĊAlternatively,ĠthinkĠofĠtotalĠcolorings:512.ĊĊNumber
Rollout 45, Act: -3.268
Ġ0,ĠbutĠlet'sĠsee.ĊĊAlternatively,ĠthinkĠaboutĠsettingĠxĠorĠyĠtoĠ0.ĠLet
Rollout 33, Act: -3.272
,ĠbutĠmaybeĠthisĠisĠnotĠstraightforward.ĠAlternatively,ĠthinkĠofĠassigningĠnumbersĠstepĠbyĠstepĠandĠmakingĠsureĠthat
Rollout 5, Act: -3.287
ĠbutĠtheseĠareĠnotĠsimilarĠnecessarily.ĠĊĊAlternatively,ĠthinkĠalgebraically.ĠLetĠmeĠdenoteĠtheĠoriginalĠtrape
Rollout 23, Act: -3.291
0Ġmin,ĠnotĠtotalĠdecays.ĊĊAlternatively,ĠthinkĠofĠtheĠproblemĠusingĠBatemanĠequationĠifĠXĠdec
Rollout 52, Act: -3.297
ĠtoĠbeĠpositive.ĠWait,ĠbutĠearlierĠreasoningĠwasĠbasedĠonĠnĠandĠkĠfromĠtheĠformulaĠwhereĠkĠis

UP_PROJ

Top 16 Positive Activations
Rollout 70, Act: 1.403
ĠinĠbase-10,ĠD=2N,ĠbutĠNĠinĠ343ĠtoĠ2
Rollout 42, Act: 1.366
ĠinĠtheĠcyclopropane,ĠformingĠaĠmoreĠstableĠcarbocationĠelsewhere.ĠSoĠifĠtheĠcyclopro
Rollout 70, Act: 1.355
ĠnumberĠD.ĠSo,ĠDĠ=Ġ2N.ĠAlso,ĠsinceĠtheĠbase-7ĠrepresentationĠofĠN
Rollout 50, Act: 1.345
log_{2x}(202x)\).ĠUsingĠtheĠchangeĠofĠbaseĠformula:ĊĊLeft-handĠside
Rollout 26, Act: 1.305
ĠatĠ298ĠK.ĠWeĠneedĠtoĠfindĠtheĠstandardĠmolarĠenthalpyĠofĠformationĠof
Rollout 42, Act: 1.303
drideĠorĠalkylĠshiftĠtoĠformĠaĠmoreĠstableĠcarbocation.ĠHowever,ĠsinceĠcyclopropane
Rollout 43, Act: 1.296
1⣩)ĠâĬĹĠ|0⣩.ĠIĠneedĠtoĠapplyĠtheĠC-NOTĠgateĠtoĠthisĠinput
Rollout 99, Act: 1.275
ĠandĠx2Ġ=Ġ1.ĠThen,ĠweĠneedĠtoĠfindĠtheĠdecimalĠvalueĠofĠx3.ĠHmm
Rollout 25, Act: 1.242
9898Ġkm,ĠandĠweĠneedĠtoĠfindĠtheĠrequiredĠmassĠsuchĠthatĠtheĠperigeeĠdistance
Rollout 23, Act: 1.183
0385ĠsâĪĴ1Ġ*N(t).ĊĊButĠwithĠN(t)=5e34Ġ*
Rollout 49, Act: 1.177
ĠÏĥ-compact,ĠthenĠXĠmustĠbeĠcountable.ĠTheĠconclusionĠisĠthatĠXĠisĠcountableĠasĠa
Rollout 49, Act: 1.169
ĠÏĥ-compact.ĠTherefore,ĠtheĠcontrapositiveĠwouldĠbe:ĠIfĠXĠisĠuncountableĠandĠHaus
Rollout 97, Act: 1.168
O4(g)ĠandĠO2(g).ĠTheĠquestionĠsaysĠit'sĠstartedĠinĠaĠclosedĠcylinderĠunderĠis
Rollout 23, Act: 1.165
Ġways:ĠalphaĠdecayĠandĠbeta-minusĠdecay.ĠTheĠquestionĠasksĠforĠtheĠactivityĠofĠtheĠalphaĠdecayĠafterĠ
Rollout 67, Act: 1.164
ĠmultiplyingĠtheĠcorrespondingĠtermsĠofĠtwoĠarithmeticĠsequences.ĠIĠneedĠtoĠfindĠtheĠeighthĠtermĠofĠthisĠsequence.ĠHmm
Rollout 95, Act: 1.161
ĠB(9,0).ĠSimilarly,ĠQkĠisĠonĠAC,ĠwhichĠgoesĠfromĠA(0,
Top 16 Negative Activations
Rollout 65, Act: -2.183
ĠaĠmustĠbeĠaĠmultipleĠofĠ1995.ĠSinceĠtheĠoriginalĠorderingĠhasĠaĠâī¤19
Rollout 65, Act: -2.186
2Ġ/1995=1995.ĠThereforeĠcĠis1995,Ġbut
Rollout 20, Act: -2.202
Ġis42.ĠTherefore,Ġ2ĠisĠaĠprimitiveĠrootĠmodulo43.ĊĊWaitĠmyĠpriorĠthoughtĠwas
Rollout 97, Act: -2.207
ĠtheĠrateĠcanĠalsoĠbeĠexpressedĠinĠtermsĠofĠpartialĠpressure.ĊĊThus,ĠtheĠrateĠofĠdecompositionĠofĠN2
Rollout 20, Act: -2.212
6Ġrespectively.ĊĊInĠcyclicĠgroups,ĠtheĠmultiplicativeĠordersĠofĠelementsĠareĠallĠtheĠdivisorsĠofĠtheĠgroup
Rollout 65, Act: -2.218
.ĊĊĠ2.ĠCuttingĠalongĠ1995-axis,ĠsayĠyĠsplitĠsuchĠthatĠsmallerĠprismĠhas
Rollout 93, Act: -2.227
om{2014}{2014}Ġ=Ġ1\),ĠweĠget:ĊĠĠĠĠ
Rollout 20, Act: -2.244
7))Ġ)=ĠLCM(1806,Ġ2162).ĠWhichĠasĠbefore
Rollout 32, Act: -2.264
KeyĠsteps:ĊĊ1.ĠIdentifyĠÏīĠisĠaĠprimitiveĠcubeĠrootĠofĠunity,ĠsoĠÏī3Ġ=
Rollout 20, Act: -2.301
ĠnĠmustĠbeĠaĠmultipleĠofĠ1806ĠandĠ2162.ĠSo,Ġn
Rollout 20, Act: -2.325
3*42/1=1806.ĊĊSimilarlyĠforĠm=47,Ġlcm(
Rollout 20, Act: -2.353
ĠmultipleĠofĠtheĠleastĠcommonĠmultipleĠofĠtheĠmultiplicativeĠordersĠofĠallĠpĠmoduloĠ43ĠandĠ4
Rollout 20, Act: -2.353
ĠbeĠtheĠleastĠcommonĠmultipleĠofĠ1806ĠandĠ2162.ĠWait,Ġbut
Rollout 20, Act: -2.388
,ĠmeaningĠthatĠtheĠleastĠcommonĠmultipleĠofĠallĠpossibleĠordersĠ(i.e.,ĠtheĠexponent)ĠisĠmĠ-
Rollout 73, Act: -2.440
.ĊĊYes,ĠtheĠformulaĠisĠr/(1Ġ-Ġr)^2Ġ-Ġ(nĠ+1)r^{n
Rollout 20, Act: -2.446
,ĠnĠmustĠbeĠLCM(1806,Ġ2162).ĠLetĠmeĠcompute

DOWN_PROJ

Top 16 Positive Activations
Rollout 57, Act: 1.958
First,Ġregularity:ĠAĠtopologicalĠspaceĠisĠregularĠif,ĠgivenĠaĠpointĠxĠandĠaĠclosedĠset
Rollout 98, Act: 1.901
ova.ĠWait,Ġno.ĠStarsĠaboveĠaroundĠ8ĠsolarĠmassesĠwillĠgoĠsupernovaĠandĠleaveĠaĠneutron
Rollout 57, Act: 1.845
ĠdisjointĠopenĠsetsĠseparatingĠxĠandĠC.ĠAlternatively,ĠregularityĠrequiresĠthatĠclosedĠsetsĠandĠpointsĠcanĠbeĠseparated
Rollout 49, Act: 1.838
ĠisĠanotherĠline.ĠXĠisĠhereditarilyĠLindelöfĠandĠT3Ġ(regularĠandĠHausd
Rollout 19, Act: 1.826
0.ĊĊWait,ĠletĠmeĠdoĠthisĠstepĠbyĠstep.ĠLet'sĠcomputeĠ\(Ġ(kĠ-Ġ5
Rollout 42, Act: 1.818
ĠtheĠcarbocationĠundergoesĠaĠWagner-MeerweinĠrearrangement.ĠIfĠtheĠinitialĠcarbocationĠisĠat
Rollout 69, Act: 1.806
Ġhere.ĊĊAlternatively,ĠsinceĠtheĠproblemĠinvolvesĠanĠinfiniteĠproductĠorĠsumĠmaybeĠinvolvingĠrecursion,ĠgivenĠtheĠdivisionĠby
Rollout 49, Act: 1.806
elöf.ĊĊInĠaĠhereditarilyĠLindelöfĠfirst-countableĠHausdorffĠspace
Rollout 49, Act: 1.792
ensationĠpoint.ĊĊWait,ĠinĠaĠLindelöfĠspace,ĠeveryĠuncountableĠsubsetĠhasĠaĠcond
Rollout 74, Act: 1.784
Ġthese.ĊĊWait,Ġlet'sĠapproachĠthisĠstepĠbyĠstep.ĠTheĠcompassĠneedleĠpointsĠinĠtheĠdirectionĠofĠthe
Rollout 98, Act: 1.774
ĠWait,Ġno.ĠStarsĠaboveĠaroundĠ8ĠsolarĠmassesĠwillĠgoĠsupernovaĠandĠleaveĠaĠneutronĠstarĠor
Rollout 57, Act: 1.770
ĠtakeĠtheĠdefinitionĠofĠregularity:ĠaĠspaceĠisĠregularĠifĠandĠonlyĠifĠforĠeveryĠpointĠxĠandĠclosed
Rollout 49, Act: 1.766
ĠuseĠtheĠfactĠthatĠXĠisĠhereditarilyĠLindelöfĠandĠHausdorff,ĠhenceĠit
Rollout 19, Act: 1.760
Ġmod83.ĠLet'sĠcomputeĠthisĠstepĠbyĠstep.ĊĊFirstĠcomputeĠ78Ġmod83=
Rollout 49, Act: 1.759
ĠanotherĠway:ĠsinceĠXĠisĠhereditarilyĠLindelöfĠandĠfirst-countable,ĠletĠmeĠassume
Rollout 49, Act: 1.754
editarilyĠLindelöfĠandĠT3Ġ(regularĠandĠHausdorff).ĠToĠshowĠthatĠX
Top 16 Negative Activations
Rollout 72, Act: -1.652
ĠâĩĴĠRe(ζ)=0.ĠWhichĠwouldĠhappenĠifĠÎĠisĠiĠorĠ-i.ĠButĠRe
Rollout 21, Act: -1.652
)=0ĠisĠsetĠforĠallĠx,ĠbutĠthatĠrestrictsĠf(x)ĠâĪĪĠ{-1,0,
Rollout 57, Act: -1.654
.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠifĠwe
Rollout 21, Act: -1.661
Ġc.ĠThenĠimageĠisĠR,ĠwhichĠdoesn'tĠfit.ĠButĠperhapsĠaddingĠanotherĠequation.ĊĊAlternatively,Ġperhaps
Rollout 82, Act: -1.669
9:Ġ0,1ĊĊButĠinĠthatĠcase,ĠcodewordsĠwouldĠbeĠ'70',
Rollout 37, Act: -1.684
Ġ$\detĠA$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠjustĠcameĠacrossĠthisĠmathĠproblem
Rollout 15, Act: -1.693
ĠoffĠafterĠsomeĠtime?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtryĠtoĠfigureĠoutĠthisĠproblem
Rollout 97, Act: -1.724
~N}_{2}Ġ\mathrm{O}_{5}(g)Ġ\stackrel{\Delta}{\
Rollout 57, Act: -1.728
ĠcontainedĠinĠtheĠcomplementĠofĠU.ĠThatĠisĠpossibleĠif,ĠforĠeveryĠyĠâĪĪĠC,ĠcomplementĠofĠU
Rollout 1, Act: -1.735
ĠsuchĠsetsĠhaveĠcertainĠproperties.ĊĊAlternatively,ĠIĠmightĠneedĠtoĠuseĠtheĠparallelogramĠlaw.ĠLetĠme
Rollout 21, Act: -1.737
)Ġ+1Ġ-x=0,ĠwhichĠonlyĠoccursĠatĠtheĠboundaries.ĠSoĠthatĠwouldĠholdĠonlyĠif
Rollout 49, Act: -1.752
ĠĠisĠcountable.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthisĠproblem
Rollout 84, Act: -1.891
/9)*(1Ġ-Ġm/n).ĠButĠthatĠleadsĠtoĠaĠdifferentĠequation.ĊĊButĠwhichĠinterpretationĠisĠcorrect
Rollout 75, Act: -1.901
ĠisĠ3ĠmodĠ4,ĠwhichĠisĠpossibleĠifĠthereĠareĠthreeĠsquaresĠthatĠareĠ1ĠmodĠ
Rollout 21, Act: -1.955
))ĠwouldĠalsoĠbeĠf(x),ĠbutĠthatĠonlyĠenforcesĠthatĠfĠisĠidempotentĠonĠitsĠimage
Rollout 21, Act: -1.960
ĠimageĠisĠaĠsubsetĠofĠZ.ĠForĠthat,ĠmaybeĠifĠweĠcanĠwriteĠanĠequationĠthatĠsaysĠf(x

Layer 22

GATE_PROJ

Top 16 Positive Activations
Rollout 1, Act: 2.851
Ġ$Ġ\mathcal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshow
Rollout 69, Act: 2.816
ĠBonanzaĠTeamĠRoundĠ#10*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfind
Rollout 58, Act: 2.813
ProposedĠbyĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠSo
Rollout 69, Act: 2.760
Ġcos(30°/3^i)]ĊĊIĠneedĠtoĠevaluateĠthatĠinfiniteĠsumĠandĠexpressĠitĠas
Rollout 18, Act: 2.758
{(7,Ġ5)}\).ĊĊ<|im_start|>answerĊToĠsolveĠtheĠproblem,ĠweĠneedĠtoĠfindĠa
Rollout 16, Act: 2.758
ProposedĠbyĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠEmiliaĠisĠtrying
Rollout 0, Act: 2.744
**Ċ\boxed{128}ĊĊ<|im_start|>answerĊGivenĠaĠrationalĠnumberĠbetweenĠ0ĠandĠ1
Rollout 28, Act: 2.742
ĠtheĠcentersĠofĠtheseĠhyperbolas.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthis
Rollout 45, Act: 2.735
ĠpossibleĠvalueĠofĠĠ$Ġm$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠproblem
Rollout 70, Act: 2.730
51Ġ=Ġ102.ĠCool.ĠIĠneedĠtoĠfindĠtheĠlargestĠsuchĠN.ĊĊFirst,
Rollout 2, Act: 2.717
109ĠinĠbothĠcases.ĠSoĠTheĠanswerĠis109.ĠHuh.ĊĊButĠwait,
Rollout 63, Act: 2.706
ĠA1ĠforĠtheĠthree-variableĠcase.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfind
Rollout 28, Act: 2.697
,ĠtheĠmidpointĠofĠOG?ĠOrĠrelatedĠtoĠmedians.ĊĊWait,ĠinĠtheĠgeneralĠcoordinateĠsystem,ĠG
Rollout 31, Act: 2.675
{5}}{2}}\).ĊĊ<|im_start|>answerĊGivenĠrealĠnumbersĠ\(ĠxĠ\)ĠandĠ\(Ġy
Rollout 68, Act: 2.671
**Ċ\boxed{501}ĊĊ<|im_start|>answerĊToĠfindĠtheĠnumberĠofĠintegerĠvaluesĠofĠ\(Ġk
Rollout 13, Act: 2.668
{187.12}ĊĊ<|im_start|>answerĊToĠdetermineĠhowĠmuchĠmoreĠmoneyĠJoanieĠwillĠowe
Top 16 Negative Activations
Rollout 3, Act: -1.249
,953.6;ĠdividedĠbyĠÏĢ*0.222ĠâīĪĠ0.
Rollout 17, Act: -1.257
.ĠSoĠsubstituting:ĊĊNumerator=2Ġ*4ÏĢe-7Ġ*1.41e
Rollout 3, Act: -1.261
ĠasĠearlierĠforĠradon)ĊĊDenominator:Ġpi*0.028=Ġ~0.0
Rollout 92, Act: -1.271
ĠfromĠlens,ĠdoĠ=Ġ20ĠcmĠ-1.5ĠcmĠ=Ġ18.5Ġcm
Rollout 81, Act: -1.300
0)ĊĊ=Ġnumerator=1020Ġ*121Ġ=1020*12
Rollout 26, Act: -1.303
s.ĊĊButĠ20.00Ġ*14.8:Ġ14.8Ġhas
Rollout 9, Act: -1.304
:Ġy=Ġ(-5/25)xĠ+50/25=Ġ(-1/5)x
Rollout 25, Act: -1.307
e20ĊĊDivideĠbyĠ(2Ġ*6.357e6Ġ)=12
Rollout 3, Act: -1.308
.314*298/(ÏĢ*0.222))ĠâīĪĠsqrt(
Rollout 56, Act: -1.310
,ĠneedĠexactĠfraction:ĊĊ1841Ġ/1008Ġsimplifies:ĠLetâĢĻsĠsee:ĊĊ
Rollout 3, Act: -1.317
.314*298/(ÏĢ*0.028))ĠâīĪĠsqrt(
Rollout 3, Act: -1.322
811.04.ĠDivideĠbyĠÏĢ*0.222âīĪ0.69
Rollout 36, Act: -1.334
,662,592Ġ÷16=23,478,9
Rollout 16, Act: -1.356
=1/2.ĠSoĠterm=6Ġ*1Ġ/Ġ(1/2)=6*2=
Rollout 17, Act: -1.391
ĠwithĠfullĠnumbers:ĊĊ|I|=(2Ġ*4ÏĢe-7Ġ*1.41e
Rollout 67, Act: -1.396
12,...ĊĊMultiplyĠthemĠterm-wise:ĊĊ8Ġ*180=1440,ĊĊ11

UP_PROJ

Top 16 Positive Activations
Rollout 64, Act: 2.579
}^8Ġ(MĠ-KyĠ-0Ġ+1Ġ)?ĠWait,ĠperhapsĠthere'sĠconfusionĠhere.ĊĊ
Rollout 46, Act: 2.497
}Ġ=Ġ-âĪĩUĠ-Ġ2ĠÏīĠÃĹĠvĠ+ĠcentrifugalĠterm.ĠWait,Ġno,
Rollout 25, Act: 2.480
Ġm,ĠviaĠmu_finalĠ=ĠG*(MĠ+Ġm).ĠButĠifĠyouĠneedĠtoĠachieveĠaĠsmallerĠper
Rollout 64, Act: 2.468
ĠsolutionsĠzĠisĠ(MĠ-KĠyĠ)Ġ+1=Ġ(2^{2012}Ġ-
Rollout 37, Act: 2.423
ĠcontributeĠaĠmultipleĠofĠn!ĠbyĠorbit-stabilizerĠtheorem.ĠTherefore,ĠtheĠwholeĠdeterminantĠisĠaĠsum
Rollout 53, Act: 2.415
ĠTherefore,ĠeachĠmedianĠlength:ĠCEĠ=ĠCPĠ+ĠPEĠ=Ġ3.0Ġ+Ġ1.5
Rollout 64, Act: 2.394
_{y=0}^8Ġ(MĠ-KyĠ-0Ġ+1Ġ)?ĠWait,ĠperhapsĠthere
Rollout 28, Act: 2.368
3)Ġ-4*(k/3)Ġ+1=0ĊĊExpand:ĊĊMultiplyĠthroughĠby9:ĊĊ
Rollout 28, Act: 2.344
aĠ)Ġ-4Ġ(kĠ/bĠ)Ġ+1=0.ĊĊMultiplyĠbothĠsidesĠbyĠa2b
Rollout 71, Act: 2.335
Ġ=Ġ-3c2Ġ+Ġ8044,ĠorĠequivalently,Ġ3c2Ġ+
Rollout 90, Act: 2.329
.ĊĊButĠlet'sĠstartĠwithĠ9NĠ+ĠMĠâī¡Ġ0ĠmodĠ1009
Rollout 64, Act: 2.319
12}Ġ-1Ġ-KĠy)Ġ+1.ĠButĠ2^{2012}
Rollout 64, Act: 2.316
}Ġ-1Ġ-KĠyĠ-0Ġ)Ġ+1=2^{2012}Ġ-K
Rollout 52, Act: 2.307
Ġ(4028/3Ġ-3Ġ+1)/2=Ġ(1342.6
Rollout 16, Act: 2.306
/63.ĠSoĠ5Ġ+11/217Ġ-1Ġ-1/63
Rollout 17, Act: 2.302
ĠTesla,Ġ**I**ĠisĠtheĠintegralĠofĠ**B**ĠtimesĠprobabilityĠdensityĠ(unitless,ĠsinceĠ|
Top 16 Negative Activations
Rollout 78, Act: -1.343
.ĠTheĠanswerĠisĠ49.ĊĊ**FinalĠAnswer**Ċ\boxed{49}ĊĊ<|im_start|>answerĊ
Rollout 55, Act: -1.347
,ĠtheĠfinalĠanswerĠisĠasĠabove.ĊĊ**FinalĠAnswer**Ċ\boxed{\begin{pmatrix}Ġ\
Rollout 70, Act: -1.348
\boxed{315}.ĊĊ**FinalĠAnswer**Ċ\boxed{315}ĊĊ<|im_start|>answer
Rollout 58, Act: -1.357
037171}.ĊĊ**FinalĠAnswer**Ċ\boxed{203717
Rollout 38, Act: -1.370
ĠisĠ\boxed{61}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{61}ĊĊ<|im_start|>answerĊ
Rollout 53, Act: -1.374
frac{27}{2}}.ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{27}{
Rollout 40, Act: -1.378
ĠisĠ\boxed{19}.ĊĊ**FinalĠAnswer**Ċ\boxed{19}ĊĊ<|im_start|>answerĊ
Rollout 13, Act: -1.383
Ġ$187.12.ĊĊ**FinalĠAnswer**Ċ\boxed{187.12
Rollout 3, Act: -1.392
{169}Ġm/s.ĊĊ**FinalĠAnswer**ĊTheĠmeanĠmolecularĠspeedĠofĠradonĠgasĠis
Rollout 25, Act: -1.395
}Ġ\text{Ġkg}}$ĊĊĊ**FinalĠAnswer**ĊTheĠasteroidĠwouldĠneedĠtoĠbeĠ$\boxed{
Rollout 4, Act: -1.406
ĠisĠapproximatelyĠ0.88%.ĊĊ**FinalĠAnswer**ĊTheĠpercentageĠerrorĠinĠtheĠcalculationĠofĠtheĠenergy
Rollout 2, Act: -1.412
ĠisĠ\boxed{109}ĊĊ**FinalĠAnswer**Ċ\boxed{109}ĊĊ<|im_start|>answer
Rollout 97, Act: -1.418
boxed{2.30}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{230}ĊĊ<|im_start|>answer
Rollout 73, Act: -1.477
,Ġ\boxed{398}ĊĊ**FinalĠAnswer**Ċ\boxed{398}ĊĊ<|im_start|>answer
Rollout 48, Act: -1.498
,ĠtheĠheaderĠdoesĠnotĠcontainĠerrors.ĊĊĊ**FinalĠAnswer**ĊTheĠIPĠheaderĠdoesĠnotĠcontainĠanyĠerrors.
Rollout 26, Act: -1.551
boxed{90.39}ĊĊĊ**FinalĠAnswer**Ċ\boxed{90.4}ĊĊ<|im_start|>

DOWN_PROJ

Top 16 Positive Activations
Rollout 93, Act: 2.102
1007ĠofĠ[cos(ÏĢk/1007)]^{2014
Rollout 93, Act: 2.021
ĠapproachĠit.ĠĊĊFirstĠoff,ĠexponentsĠwithĠcosineĠcanĠsometimesĠbeĠdealtĠwithĠusingĠpower-reductionĠformulas
Rollout 66, Act: 2.012
**:ĠTheĠtriangleĠ\(ABC_0\)ĠisĠaĠprimitiveĠPythagoreanĠtriple,ĠmeaningĠitsĠsidesĠareĠof
Rollout 66, Act: 2.011
ĠbeĠpairwiseĠcoprime,Ġit'sĠlikelyĠaĠprimitiveĠtripleĠwhereĠtheĠlegsĠareĠm2Ġ-Ġn2
Rollout 93, Act: 2.009
007.ĠSoĠ[cos(ÏĢk/1007)]^{2*10
Rollout 66, Act: 1.971
ĠisĠtoĠfindĠintegersĠa,Ġb,ĠcĠ(primitiveĠPythagoreanĠtriple)ĠsuchĠthatĠabĠ=Ġ
Rollout 66, Act: 1.967
=170ĊĊCheckĠifĠthisĠisĠaĠprimitiveĠPythagoreanĠtriple:Ġgcd(a,ĠbĠ)
Rollout 32, Act: 1.920
ĠletĠmeĠrecallĠthatĠtheĠrootsĠofĠx3Ġ=Ġ1ĠareĠ1,ĠÏī,ĠandĠÏ
Rollout 45, Act: 1.915
ĠpossibleĠvalueĠofĠĠ$Ġm$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠproblem
Rollout 7, Act: 1.914
71N.ĊĊWhichĠsimplifiesĠasĠweĠdidĠbeforeĠtoĠx=N/12.ĊĊHowever,Ġafter
Rollout 66, Act: 1.909
ĠbeĠeven.ĠSoĠanyway,ĠtheĠsidesĠformĠaĠprimitiveĠtriple.ĊĊC1ĠisĠtheĠfootĠofĠtheĠaltitude
Rollout 85, Act: 1.908
0,2024)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'mĠtryingĠto
Rollout 66, Act: 1.901
ĠsidesĠareĠpairwiseĠcoprime,Ġit'sĠaĠprimitiveĠPythagoreanĠtriple.ĠSoĠaĠ=Ġm2
Rollout 30, Act: 1.896
n+3}}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'veĠgotĠthis
Rollout 7, Act: 1.887
-promotedĠĠ$47$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'mĠtryingĠto
Rollout 93, Act: 1.859
1007}Ġ[cos(ÏĢk/1007)]^{2014
Top 16 Negative Activations
Rollout 18, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊConsiderĠtheĠadditiveĠgroup
Rollout 19, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊFindĠtheĠsumĠof
Rollout 31, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊGivenĠareĠrealĠnumbers
Rollout 32, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠ$\omega$
Rollout 40, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠparticleĠisĠlocated
Rollout 50, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠpositive
Rollout 35, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠ$P(x
Rollout 61, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠ$S$
Rollout 59, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊJackieĠandĠPhil
Rollout 67, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊFindĠtheĠeighthĠterm
Rollout 58, Act: -1.727
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊPatchouliĠis
Rollout 88, Act: -1.731
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊConsiderĠConvolutional
Rollout 4, Act: -1.738
1/C_eqĠ=Ġ1/C1Ġ+Ġ1/C2.ĠSo,ĠIĠcanĠwriteĠthatĠas
Rollout 92, Act: -1.841
:Ġ1/fĠ=Ġ1/doĠ+Ġ1/di.ĠHere,ĠdoĠisĠtheĠobjectĠdistance
Rollout 4, Act: -1.845
/C_eqĠ=Ġ1/C1Ġ+Ġ1/C2.ĠSo,ĠIĠcanĠwriteĠthatĠasĠC
Rollout 8, Act: -1.901
ĠtheĠnumberĠofĠsequencesĠisĠtheĠnumberĠofĠinjectiveĠfunctionsĠfromĠ5ĠpositionsĠtoĠ10Ġcards,

Layer 23

GATE_PROJ

Top 16 Positive Activations
Rollout 69, Act: 1.621
ĠleftĠside:ĊĊ=2âĪļ3Ġ*aĠ+2âĪļ3Ġ*âĪļbĠ-3aĠ-3
Rollout 96, Act: 1.608
132Ġ+Ġ12y)Ġ+25yĠ=450ĊĊ12*
Rollout 36, Act: 1.604
*1,800*56Ġ+562=3,240,0
Rollout 81, Act: 1.495
x2Ġ+298xĠ+9*289=x2Ġ+298xĠ+
Rollout 69, Act: 1.411
3Ġ-3Ġ)Ġ/12Ġ=Ġ(2*3Ġ-3âĪļ3Ġ)Ġ/12âĪļ
Rollout 38, Act: 1.406
=(6*(24/7Ġx)Ġ+37x)/31=(144/
Rollout 28, Act: 1.404
)^2Ġ=Ġh2Ġ+Ġ2hkĠ+k2Ġ-4hĠ-4kĠ+4ĊĊ
Rollout 28, Act: 1.391
Ġ+k2Ġ-4hĠ-4kĠ+4ĊĊThen,ĠexpandĠ(hĠ-k)^2
Rollout 69, Act: 1.374
ĠlikeĠterms:ĊĊ=Ġ(2aâĪļ3Ġ+2âĪļ3Ġ*âĪļbĠ)Ġ-3aĠ-
Rollout 4, Act: 1.370
ĠinĠC_eq:Ġsqrt(Ġ(Ġ(10*3000^2)/(200
Rollout 36, Act: 1.367
Expand:ĊĊx2Ġ+Ġx2Ġ+Ġ(2Ġ*ĠxĠ*Ġ9000/s)Ġ+
Rollout 64, Act: 1.360
2009}-1Ġ)ĊĊ=Ġ[8*2^{2009}+a1
Rollout 69, Act: 1.351
2Ġ-âĪļ3Ġ)âĪļ3Ġ)/(4*3Ġ)=Ġ(2âĪļ3Ġ-3Ġ)
Rollout 95, Act: 1.343
3(2k-1)/Ġ(2*2450Ġ)ĊĊĠDistanceĠbetweenĠisĠ(9Ġsqrt
Rollout 31, Act: 1.339
2]/4=(1Ġ+2âĪļ5Ġ+5)/4=(6Ġ+2âĪļ5)/4
Rollout 4, Act: 1.329
5Ġ-10*3000Ġ+10*15=6,000
Top 16 Negative Activations
Rollout 56, Act: -3.140
ĠinĠstandardĠ30-60-90Ġtriangles,ĠtheĠsidesĠareĠ1Ġ(opposite
Rollout 10, Act: -3.145
7-16=1,ĠwhichĠisĠaĠwinningĠpositionĠforĠopponent.ĠSoĠallĠmovesĠfromĠn=
Rollout 37, Act: -3.147
2.Ġ3!=6,ĠwhichĠdoesĠnotĠdivideĠ2.ĠHence,Ġ2015
Rollout 20, Act: -3.148
)Ġâī¡Ġ1ĠmodĠ2021.ĠSinceĠthisĠhasĠtoĠholdĠforĠallĠa,
Rollout 67, Act: -3.153
80ĠandĠ24.ĠTheĠGCDĠofĠ180ĠandĠ24ĠisĠ
Rollout 15, Act: -3.156
]],ĠwhichĠisĠidentity.ĠTherefore,ĠnotĠnilpotent.ĠForĠn=3,ĠasĠseenĠearlier,
Rollout 2, Act: -3.158
Ġ2^3Ġ=Ġ8Ġâī¡Ġ0ĠmodĠ8,ĠandĠ2^kĠfor
Rollout 27, Act: -3.178
.491e5Ġm/s.ĊĊYes,ĠsoĠapproximatelyĠ1.491ĠÃĹĠ
Rollout 62, Act: -3.200
*47=Ġ517.ĠYes,Ġ11*47.ĠSoĠtwoĠprimes
Rollout 16, Act: -3.202
=1*2,ĠsoĠgcdĠisĠ1.ĠHence,Ġ7880/19
Rollout 62, Act: -3.209
11*165.2Ġ->Ġnot.ĠByĠ13:13*13
Rollout 37, Act: -3.217
.Ġ3!Ġ=6,ĠwhichĠdoesĠnotĠdivideĠ2.ĠSoĠn!ĠisĠnotĠaĠfactor
Rollout 23, Act: -3.221
.51ĠBq.ĠHmm,ĠstillĠclosestĠtoĠoptionĠBĠ(3.719ĠBq
Rollout 90, Act: -3.247
,396=448,Ġ0?ĠLet'sĠcomputeĠ403,6
Rollout 96, Act: -3.261
3:Ġ4+1=5,ĠnotĠdivisibleĠbyĠ3.ĠHenceĠno.ĠSoĠdenominatorĠisn
Rollout 81, Act: -3.299
^2=29584.ĠYes.ĠSoĠsqrt(29584)=1

UP_PROJ

Top 16 Positive Activations
Rollout 49, Act: 2.418
?ĠLetĠmeĠcheck.ĊĊTakeĠanĠuncountableĠdiscreteĠspaceĠD,ĠandĠaddĠaĠpointĠatĠinfinity,
Rollout 49, Act: 2.397
ĠyouĠtakeĠaĠsubsetĠthatĠisĠuncountableĠandĠdiscrete,ĠitĠwouldĠnotĠbeĠLindelöf.ĊĊ
Rollout 49, Act: 2.389
countableĠandĠdiscrete.ĠButĠanĠuncountableĠdiscreteĠspaceĠisĠnotĠLindelöf,ĠsoĠthis
Rollout 49, Act: 2.369
ĠtheĠone-pointĠcompactificationĠofĠanĠuncountableĠdiscreteĠspaceĠisĠnotĠhereditarilyĠLindelöf
Rollout 49, Act: 2.314
able,ĠmaybeĠweĠcanĠconstructĠanĠuncountableĠdiscreteĠsubspace?ĊĊButĠinĠhereditarilyĠLindel
Rollout 49, Act: 2.308
ĠaĠsubspace,ĠwhichĠisĠuncountableĠandĠdiscrete.ĠButĠanĠuncountableĠdiscreteĠspaceĠisĠnot
Rollout 49, Act: 2.261
ĠtheĠone-pointĠcompactificationĠofĠanĠuncountableĠdiscreteĠspaceĠisĠcompact,ĠhenceĠLindelöf,
Rollout 49, Act: 2.258
f,ĠthenĠitĠcannotĠcontainĠanĠuncountableĠdiscreteĠsubspace.ĠButĠifĠXĠisĠuncountable
Rollout 49, Act: 2.249
ĠtheĠCantor-BendixsonĠdecompositionĠisĠaĠdiscreteĠsubspace.ĠForĠLindelöf,Ġall
Rollout 49, Act: 2.232
ĠtheĠone-pointĠcompactificationĠofĠanĠuncountableĠdiscreteĠspaceĠisĠscatteredĠbutĠnotĠhereditarilyĠLindel
Rollout 49, Act: 2.205
Ġthen,ĠinĠaĠLindelöfĠHausdorffĠspace,ĠanyĠclosedĠsetĠisĠalsoĠLindel
Rollout 49, Act: 2.143
ĠinĠthisĠspace,ĠthereĠareĠnoĠuncountableĠdiscreteĠsubspaces,ĠsoĠXĠisĠaĠcontinuousĠimageĠof
Rollout 49, Act: 2.133
.ĠThenĠAĠmustĠcontainĠaĠcountableĠunionĠofĠcompactĠsets.ĊĊEachĠcompactĠsubsetĠinĠaĠHausdor
Rollout 49, Act: 2.131
ĠhereditarilyĠLindelöfĠspaces,ĠeveryĠdiscreteĠsubspaceĠisĠcountable.ĠSoĠifĠXĠis
Rollout 49, Act: 2.109
or-BendixsonĠdecompositionĠisĠaĠdiscreteĠsubspace.ĠForĠLindelöf,ĠallĠdiscreteĠsub
Rollout 49, Act: 2.107
ĠaĠhereditarilyĠLindelöfĠHausdorffĠspaceĠisĠregular.ĠLet'sĠsee:ĠTake
Top 16 Negative Activations
Rollout 62, Act: -2.041
ĠthatĠexpressionĠisĠ11*q*q*p,ĠtotalĠfourĠprimes.ĊĊSoĠpossible.ĊĊSoĠtestingĠthisĠcase
Rollout 56, Act: -2.045
âĪļ3Ġ/144ĊĊSumĠbetweenĠbrackets:Ġ(263âĪļ3Ġ/144
Rollout 62, Act: -2.057
11*(13*31),ĠsoĠtotalĠthreeĠprimeĠfactors.ĊĊSoĠforĠm=11
Rollout 62, Act: -2.062
1)=a*b*c,ĠthreeĠprimes.ĠThenĠtotalĠprimeĠfactorsĠwouldĠbeĠ11*a*b*c
Rollout 75, Act: -2.063
2Ġ+Ġ(2d)^2.ĠThenĠtotalĠexpression:4pĠ+3=Ġ(2a)^
Rollout 62, Act: -2.092
12ĠgivesĠ11*13Ġ(totalĠtwoĠprimes).ĊĊk=2:11*(
Rollout 89, Act: -2.104
66ĠandĠboysĠ100.ĠThenĠfractionĠofĠgirlsĠisĠ66/166
Rollout 71, Act: -2.107
:202=400.ĠthenĠleftĠ4022Ġ-400=
Rollout 30, Act: -2.113
8,210.ĊĊTherefore,ĠtheĠreducedĠformĠwouldĠbeĠ4058211
Rollout 45, Act: -2.120
ĠâĨĴĠ(nĠ+c)x.ĠCompareĠleftĠwhichĠhasĠnĠx.ĠTherefore:ĊĊnĠ=ĠnĠ+c
Rollout 71, Act: -2.139
2=3969.ĠThenĠremainingĠsumĠis4022Ġ-3969=
Rollout 71, Act: -2.161
32=3969.ĠThenĠremainingĠsumĠis4022Ġ-3969
Rollout 70, Act: -2.161
57,ĠsoĠnextĠdigitĠ6.ĠFinallyĠremainderĠ48-6*7=6.
Rollout 62, Act: -2.196
*13*41.ĠThreeĠprimes.ĠTotalĠthreeĠprimes.ĊĊStillĠinsufficient.ĊĊm=88
Rollout 45, Act: -2.207
ĠxĠâĨĴĠ(nĠ+c)x.ĠCompareĠleftĠwhichĠhasĠnĠx.ĠTherefore:ĊĊnĠ=ĠnĠ+
Rollout 71, Act: -2.348
202=400.ĠthenĠleftĠ4022Ġ-400=3

DOWN_PROJ

Top 16 Positive Activations
Rollout 65, Act: 3.273
Ġsquare,ĠsoĠprimesĠinĠitsĠfactorizationĠhaveĠevenĠexponents.ĠTherefore,ĠsinceĠ1995
Rollout 65, Act: 2.691
Ġis,ĠifĠa2ĠhasĠatĠminimumĠtheĠexponentsĠ1ĠinĠeachĠprime,ĠnoĠmatterĠhow,
Rollout 0, Act: 2.664
ĠkĠisĠprimes.ĠBut,ĠsinceĠprimeĠfactorsĠwithĠexponents:ĠWait,ĠbutĠeachĠprimeĠisĠeitherĠincluded
Rollout 65, Act: 2.515
ĠatĠleastĠ1Ġ(sinceĠifĠaĠprimeĠhasĠexponentĠ1ĠinĠa,ĠthenĠexponentĠ2Ġin
Rollout 65, Act: 2.461
995ĠisĠsquare-freeĠ(allĠitsĠprimeĠexponentsĠare1),ĠthenĠaĠmustĠequalsĠ1
Rollout 37, Act: 2.422
ĠbyĠallĠprimesĠupĠtoĠnĠbutĠnotĠtheirĠmultiplicities.ĠForĠexample,Ġ4ĠdividesĠ2*
Rollout 52, Act: 2.381
ĠdivisorsĠisĠ(2+1)*(1+1)*(1+1)=Ġ3*2*
Rollout 0, Act: 2.376
ĠfactorizationĠofĠ20!.ĊĊToĠcomputeĠtheĠexponentsĠofĠprimesĠinĠ20!:ĊĊFor
Rollout 65, Act: 2.344
ĠdividesĠa2ĠonlyĠifĠinĠa2,ĠtheĠexponentsĠforĠ3,5,7,1
Rollout 0, Act: 2.334
ĠnotĠaĠsquareĠ(unlessĠforĠexampleĠprimesĠwithĠevenĠexponents),ĠitĠdepends.ĠAsĠwithĠN!Ġfor
Rollout 65, Act: 2.325
ĠsituationĠis,ĠifĠa2ĠhasĠatĠminimumĠtheĠexponentsĠ1ĠinĠeachĠprime,ĠnoĠmatterĠhow
Rollout 20, Act: 2.321
.ĠToĠcheck:ĊĊTheĠLCMĠtakesĠtheĠmaxĠexponentsĠofĠeachĠprime:ĊĊPrimesĠinvolved:Ġ
Rollout 65, Act: 2.299
*5*7*19,ĠeachĠwithĠexponentĠ1.ĠSoĠ1995Ġdivides
Rollout 65, Act: 2.260
*7*19.ĠTherefore,ĠprimesĠwithĠexponentĠodd.ĠNeedĠ"a"ĠtoĠbeĠmultipliedĠ
Rollout 65, Act: 2.240
92.ĠNumberĠofĠdivisorsĠisĠ(2+1)(2+1)(2+1)(
Rollout 37, Act: 2.206
015!ĠisĠtheĠproductĠp1^e1Ġ*p2^e2Ġ*...*
Top 16 Negative Activations
Rollout 34, Act: -2.057
(s)\}ds$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠtheseĠtwoĠintegralĠequations
Rollout 37, Act: -2.068
Ġ$\detĠA$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠjustĠcameĠacrossĠthisĠmathĠproblem
Rollout 58, Act: -2.096
ĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠSoĠPatchouli
Rollout 93, Act: -2.099
007}ĠforĠN=1008:ĠNo,Ġwait,ĠtheĠangleĠwasĠ2
Rollout 11, Act: -2.104
*DavidĠAltizio*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠproblemĠhereĠwhereĠa
Rollout 16, Act: -2.107
ĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠEmiliaĠisĠtryingĠtoĠgrowĠsix
Rollout 49, Act: -2.113
ĠĠisĠcountable.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthisĠproblem
Rollout 12, Act: -2.113
P.ĠTuran]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠproveĠthatĠtheĠfunction
Rollout 33, Act: -2.114
3,4),(3,5),(4,5).ĊĊComputeĠeach:ĊĊ(1,2):Ġ
Rollout 21, Act: -2.170
childkraut*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠpartĠ(a)ĠisĠaskingĠto
Rollout 37, Act: -2.193
ĠanyĠfunctionĠf_xĠisĠperiodicĠwithĠperiodĠdividingĠp-1.ĠTherefore,ĠthereĠareĠatĠmostĠp-
Rollout 72, Act: -2.204
ĠĊ1.Ġ$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblem:ĠFindĠall
Rollout 93, Act: -2.210
2^{2m}=4^{m}.ĠThenĠC(2m,m)=4^mĠ(sqrt
Rollout 69, Act: -2.263
ĠRoundĠ#10*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠpositiveĠintegersĠa
Rollout 28, Act: -2.361
ĠtheseĠhyperbolas.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthisĠproblemĠhereĠabout
Rollout 20, Act: -2.401
ĠTheĠexponentĠofĠaĠcyclicĠgroupĠisĠequalĠtoĠtheĠorderĠofĠtheĠgroup.ĠTherefore,ĠforĠaĠprimeĠm

Layer 24

GATE_PROJ

Top 16 Positive Activations
Rollout 33, Act: 1.557
ĠstartsĠwithĠ2Ġor3.ĠThen,ĠforĠk=2:ĠfirstĠ2ĠelementsĠmustĠcontainĠat
Rollout 20, Act: 1.447
ĠisĠcyclic,ĠthereĠexistĠprimitiveĠrootsĠwhereĠord_m(p)Ġ=ĠmĠ-1,ĠandĠelementsĠofĠother
Rollout 33, Act: 1.432
,ĠitĠcan'tĠbeĠ1.ĠThenĠforĠk=2,ĠinĠtheĠfirstĠtwoĠelements,Ġthere
Rollout 26, Act: 1.424
8Ġ\mathrm{~K}$ĠandĠ1ĠatmĠintoĠ$\mathrm{HgO}(s)
Rollout 33, Act: 1.422
],Ġ[2,1].ĠTheĠconditionĠisĠforĠk=1:ĠfirstĠelementĠ>1Ġ(soĠinvalid
Rollout 33, Act: 1.413
3ĠincludeĠnumbers.ĠHoweverĠstartingĠwith4ĠwillĠsatisfyĠk=1Ġ(notĠstartedĠwith1)ĠandĠk
Rollout 44, Act: 1.394
^(2/3).ĠButĠT2=Ġ(p2ĠV2)/(nR).ĠWait,Ġsince
Rollout 22, Act: 1.374
73.15ĠK)ĠandĠ1ĠatmĠpressure.ĠAlthoughĠpressureĠmightĠnotĠmatterĠhereĠbecauseĠthe
Rollout 20, Act: 1.364
ĠnecessaryĠthatĠknĠisĠaĠmultipleĠofĠtheĠorderĠofĠpĠmoduloĠm.ĠSinceĠ(Z/mZ)^ÃĹ
Rollout 33, Act: 1.335
>2.ĠTherefore,ĠtheĠonlyĠinvalidĠpermutationsĠforĠk=2ĠareĠ[2,1,*],Ġi
Rollout 3, Act: 1.335
ĠPressure),ĠwhichĠisĠ0°CĠandĠ1Ġatm.ĠHowever,ĠsometimesĠpeopleĠuseĠ25°C
Rollout 33, Act: 1.311
ĠisĠ3.ĠTherefore,ĠtheĠforbiddenĠonesĠforĠk=2ĠareĠfirstĠtwoĠelementsĠ{1,2
Rollout 33, Act: 1.310
Ġthere'sĠ5ĠchoicesĠhere.ĊĊThen,ĠforĠk=2,ĠtheĠfirstĠtwoĠpositionsĠmustĠnotĠboth
Rollout 20, Act: 1.302
ĠresiduesĠmodĠm,ĠweĠhaveĠthatĠtheĠorderĠofĠpĠdividesĠkn.ĠTherefore,ĠsinceĠtheĠordersĠvary,
Rollout 33, Act: 1.289
ĠisĠwhenĠtheĠfirstĠelementĠisĠ1.ĠForĠk=2,ĠtheĠforbiddenĠpermutationsĠareĠthoseĠstartingĠwith
Rollout 44, Act: 1.288
ĠBecauseĠtheĠgasĠdoesĠworkĠagainstĠtheĠhighestĠpossibleĠopposingĠpressure.ĠConversely,ĠforĠcompression,ĠtheĠmaximumĠworkĠis
Top 16 Negative Activations
Rollout 65, Act: -3.404
Ġproportion.ĠHmmm.ĊĊWait,ĠperhapsĠweĠneedĠthatĠafterĠcuttingĠaĠprismĠbyĠaĠplaneĠparallelĠtoĠone
Rollout 62, Act: -3.414
ĠmultipleĠtimes.ĠĊĊFirst,ĠmaybeĠIĠshouldĠstartĠbyĠunderstandingĠwhatĠtheĠquestionĠisĠasking.ĠIfĠweĠhave
Rollout 69, Act: -3.417
Ġfunction.ĊĊWait,ĠanotherĠidea.ĠLetĠmeĠnoteĠthatĠ3Ġtheta_iĠ=Ġtheta_{iĠ-1
Rollout 50, Act: -3.420
ĠmaybeĠthisĠisn'tĠhelpful.ĠAlternatively,ĠperhapsĠnoticeĠthatĠ101ĠandĠ11ĠareĠprimes
Rollout 11, Act: -3.420
ĠMaybeĠconsideringĠalgebraĠinsteadĠofĠanalysis.ĠSupposeĠweĠnoteĠthatĠgivenĠpĠandĠqĠareĠreal,ĠcanĠweĠrelate
Rollout 24, Act: -3.426
ĠrelyingĠonĠsomeĠunderlyingĠstructure.ĊĊPerhapsĠtheĠkeyĠisĠthatĠinĠaĠminimalĠsaddleĠpairĠ(R,ĠC),
Rollout 50, Act: -3.441
log_{20x}(22x)\),ĠwhichĠisĠequalĠtoĠ\(\log_{2x
Rollout 69, Act: -3.457
Ċ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠpositiveĠintegersĠaĠandĠbĠsuchĠthatĠ1Ġover
Rollout 50, Act: -3.466
ĠmeĠrecallĠsomeĠlogarithmicĠidentitiesĠthatĠmightĠhelpĠhere.ĠTheĠchangeĠofĠbaseĠformulaĠisĠusefulĠforĠcomparingĠlogarith
Rollout 89, Act: -3.486
ĠgirlsĠthereĠare.ĊĊSoĠtheĠtrickĠhereĠisĠprobablyĠtoĠfigureĠoutĠtheĠtotalĠnumberĠofĠchildrenĠfirst,Ġright
Rollout 41, Act: -3.542
:ĠForĠthisĠproblem,ĠtheĠuserĠisĠlikelyĠexpectedĠtoĠuseĠtheĠcasesĠbyĠnumberĠlengthĠandĠsumĠresultĠ(
Rollout 66, Act: -3.562
.ĠSo,ĠmaybeĠtheseĠlengthsĠformĠaĠgeometricĠseriesĠthatĠsumsĠtoĠ6p.ĠWeĠhaveĠtoĠfind
Rollout 50, Act: -3.653
10}20\)ĊĊLetĠmeĠnoteĠthatĠ202ĠisĠ2*10
Rollout 11, Act: -3.688
ĠisĠ1.ĠĊĊFirst,ĠletĠmeĠnoteĠthatĠsinceĠabcĠ=Ġ1,ĠmaybeĠweĠcanĠwrite
Rollout 30, Act: -3.707
Ġterm.ĠTherefore,ĠtoĠtelescope,ĠweĠmightĠneedĠthatĠ-b_nĠ+Ġb_{n+2}
Rollout 89, Act: -3.838
ĠisĠcorrect.ĊĊWait,ĠbutĠmaybeĠtheĠkeyĠisĠthatĠgivenĠboysĠareĠ100,ĠandĠgirls

UP_PROJ

Top 16 Positive Activations
Rollout 20, Act: 3.174
ĠtoĠhaveĠp^{kn}Ġâī¡Ġ1ĠmodĠmĠforĠallĠpĠnotĠdivisibleĠbyĠm,Ġit
Rollout 20, Act: 3.134
ĠweĠneedĠp^{n}Ġâī¡Ġ1ĠmodĠ43.ĠForĠkĠ=Ġ2,
Rollout 20, Act: 3.105
ĠsuchĠthatĠp^{n}Ġâī¡Ġ1ĠmodĠ43ĠandĠ47ĠforĠallĠprimes
Rollout 20, Act: 3.067
3,Ġp^{kn}Ġâī¡Ġ1ĠmodĠ43.ĠSimilarly,ĠforĠmodulusĠ4
Rollout 20, Act: 3.062
.ĊĊTherefore,ĠweĠneedĠknĠâī¡Ġ0ĠmodĠmĠ-Ġ1,ĠforĠmĠ=Ġ4
Rollout 20, Act: 3.013
ĠreducesĠtoĠp^{kn}Ġâī¡Ġ1ĠmodĠ43ĠforĠallĠprimesĠpĠâīłĠ4
Rollout 20, Act: 3.011
ĠtoĠhaveĠp^{n}Ġâī¡Ġ1ĠmodĠ43ĠforĠallĠpĠnotĠ43
Rollout 20, Act: 2.976
ĠthingĠisĠp^{kn}Ġâī¡Ġ1ĠmodĠm.ĠHmm.ĠButĠhowĠdidĠkĠfactorĠinto
Rollout 20, Act: 2.968
Ġp^{kn}Ġ\equivĠ1Ġ\pmod{43}Ġ\).ĠThisĠrequiresĠ\(Ġn
Rollout 20, Act: 2.968
ĠweĠneedĠp^{kn}Ġâī¡Ġ1ĠmodĠ47ĠforĠallĠprimesĠpĠâīłĠ4
Rollout 20, Act: 2.957
Ġp^{kn}Ġ\equivĠ1Ġ\pmod{47}Ġ\).ĠThisĠrequiresĠ\(Ġn
Rollout 20, Act: 2.949
Ġ43,Ġp^nĠâī¡Ġ1ĠmodĠ43.ĠThen,ĠforĠanyĠprimeĠp
Rollout 61, Act: 2.947
2^{kĠ-Ġm})Ġâī¡Ġ0ĠmodĠ9.ĠSinceĠ2ĠandĠ9Ġare
Rollout 20, Act: 2.941
ĠLet'sĠsupposeĠthatĠp^nĠâī¡Ġ1ĠmodĠ43.ĠThen,Ġp^{kn}
Rollout 20, Act: 2.918
ĠneedĠp^{2n}Ġâī¡Ġ1ĠmodĠ43.ĠHowever,ĠbutĠsinceĠifĠp
Rollout 20, Act: 2.905
=43:ĠneedĠknĠâī¡Ġ0ĠmodĠ42,ĠandĠm=47:
Top 16 Negative Activations
Rollout 42, Act: -1.834
ĠHere,ĠtheĠleavingĠgroupĠ(water)ĠhasĠalreadyĠleft,ĠformingĠaĠcarbocation.ĠThen,ĠtheĠbase
Rollout 42, Act: -1.841
ĠbondĠwouldĠformĠwithoutĠrearrangement,ĠbutĠgivenĠtheĠangleĠrequiredĠforĠE2,ĠwhichĠisĠtypicallyĠ1
Rollout 31, Act: -1.842
6Ġ-2âĪļ5)/4=(3Ġ-âĪļ5)/2ĊĊTherefore,Ġy=2Ġ-
Rollout 31, Act: -1.847
Ġ+âĪļ5)/2ĠandĠy=(1Ġ-âĪļ5)/2:ĊĊMultiplyĠxĠandĠy:ĊĊ(
Rollout 29, Act: -1.857
ĠlookingĠbackĠatĠtheĠoptions,ĠoptionĠEĠisĠPĠ<ĠrhoĠgĠH.ĠButĠaccordingĠtoĠtheĠcalculation,
Rollout 29, Act: -1.877
$ĠC)ĠĠ$\rhoĠgH<ĠPĠ<Ġ\rhoĠgh$ĠD)ĠĠ$PĠ>
Rollout 42, Act: -1.915
ĠnotĠbeĠveryĠstable.ĠTherefore,ĠsomeĠcarbocationĠrearrangementĠmightĠoccur.ĠForĠexample,ĠmaybeĠtheĠcyc
Rollout 42, Act: -1.916
,ĠandĠaĠhydrogen.ĠTherefore,ĠwhenĠtheĠwaterĠleavesĠ(afterĠprotonation),ĠweĠneedĠtoĠeliminateĠa
Rollout 42, Act: -1.938
Ġring,ĠitĠmightĠleadĠtoĠringĠexpansionĠorĠotherĠrearrangementsĠtoĠrelieveĠstrain.ĠWait,ĠespeciallyĠthe
Rollout 46, Act: -1.955
here,ĠpointingĠalongĠtheĠz-axisĠifĠrotationĠisĠinĠtheĠx-yĠplane),ĠsoĠtheĠcrossĠproductĠÏ
Rollout 54, Act: -1.991
icĠbecauseĠstericĠhindranceĠwouldĠpreventĠitĠfromĠapproachingĠtheĠelectrophilicĠcenterĠasĠeasily.ĠSo
Rollout 54, Act: -2.100
ĠSN2Ġreactions,ĠbulkinessĠhindersĠtheĠbacksideĠattack,ĠmakingĠitĠlessĠeffective.ĠSoĠmaybeĠcyc
Rollout 29, Act: -2.136
ĠgHĊĊC)ĠrhoĠgH<ĠPĠ<ĠrhoĠghĊĊD)ĠPĠ>ĠrhoĠghĊĊ
Rollout 29, Act: -2.229
ĠgH$ĠC)ĠĠ$\rhoĠgH<ĠPĠ<Ġ\rhoĠgh$ĠD)ĠĠ$
Rollout 54, Act: -2.370
InĠSN2Ġreactions,ĠbulkinessĠhindersĠtheĠbacksideĠattack,ĠmakingĠitĠlessĠeffective.ĠSoĠmaybe
Rollout 29, Act: -2.388
Ġ=ĠrhoĠgHĊĊC)ĠrhoĠgH<ĠPĠ<ĠrhoĠghĊĊD)ĠPĠ>Ġrho

DOWN_PROJ

Top 16 Positive Activations
Rollout 74, Act: 2.370
ĠButĠinĠtheĠproblemĠstatement,ĠitĠjustĠsaysĠ"deriveĠanĠexpressionĠforĠtheĠdeviationĠofĠtheĠcompassĠδθ
Rollout 20, Act: 2.348
Ġm=43:ĠneedĠknĠâī¡Ġ0ĠmodĠ42,ĠandĠm=47
Rollout 42, Act: 2.238
.ĊĊWait,ĠwhenĠtheĠwaterĠleaves,ĠtheĠcarbocationĠwouldĠformĠonĠtheĠcentralĠcarbon.ĠHowever,Ġhaving
Rollout 2, Act: 2.161
0Ġmod8ĊĊThus,Ġ5mâī¡0Ġmod8.ĠSoĠ5mĠisĠdivisibleĠby
Rollout 21, Act: 2.120
ĠSchildkraut*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠpartĠ(a)ĠisĠasking
Rollout 18, Act: 2.116
ĠinverseĠofĠ3Ġmodulo7,ĠwhichĠisĠ5Ġ(sinceĠ3*5=15âī¡
Rollout 20, Act: 2.113
1.ĊĊTherefore,ĠweĠneedĠknĠâī¡Ġ0ĠmodĠmĠ-Ġ1,ĠforĠmĠ=Ġ
Rollout 20, Act: 2.110
ĠpĠâī¡1ĠmodĠm,ĠnkĠâī¡0ĠmodĠm.ĠButĠhere,Ġwait.ĠWait,
Rollout 30, Act: 2.093
10.ĊĊButĠgivenĠtheĠproblemĠasksĠtoĠ"evaluateĠtheĠinfiniteĠsum",ĠlikelyĠasĠaĠfraction.ĠBut
Rollout 20, Act: 2.070
ĠpositiveĠinteger.ĠButĠifĠweĠneedĠknĠâī¡0ĠmodĠmĠforĠallĠk,ĠthenĠnĠmustĠâī
Rollout 20, Act: 2.057
Ġ-Ġ1.ĠTherefore,ĠknĠâī¡Ġ0ĠmodĠ(mĠ-Ġ1)ĠforĠeachĠmodulus
Rollout 56, Act: 2.050
ĠatĠleastĠvisualizeĠit.ĠA0ĠisĠtheĠrightĠangle,ĠA1ĠisĠ60°,ĠA
Rollout 25, Act: 2.024
$ĠisĠtheĠimpactĠparameter.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠthisĠproblem
Rollout 20, Act: 2.022
ĠifĠweĠtakeĠknĠsuchĠthatĠknĠâī¡Ġ0ĠmodĠ(mĠ-Ġ1),ĠwhichĠwouldĠbe
Rollout 7, Act: 1.991
,ĠbutĠreadingĠtheĠproblemĠagain:Ġ"dueĠtoĠanĠerrorĠinĠtheĠwordingĠofĠaĠquestion,ĠallĠscores
Rollout 57, Act: 1.975
*A.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠif
Top 16 Negative Activations
Rollout 50, Act: -2.199
ĠwhichĠtheyĠareĠ(sinceĠ101ĠisĠprimeĠandĠ11Ġ<Ġ101Ġand
Rollout 49, Act: -2.225
Ġ(asĠabove):ĊĊLetĠX'Ġ=ĠXĠwithoutĠisolatedĠpoints.ĠIfĠX'ĠisĠnon-empty,Ġthen
Rollout 50, Act: -2.227
ĠthatĠ101ĠandĠ11ĠareĠprimes,ĠandĠlikelyĠwon'tĠsimplify.ĠPerhapsĠIĠneed
Rollout 14, Act: -2.228
2}ĠdxĠisĠdivergentĠatĠbothĠ0ĠandĠâĪŀ.ĠHenceĠthisĠtermĠisĠinfinity.ĠThus
Rollout 19, Act: -2.239
ĠorĠequalĠto6.ĠSinceĠ83ĠisĠprime,ĠitsĠdivisorsĠareĠ1Ġand83
Rollout 59, Act: -2.252
CD(246,784).ĊĊPrimeĠfactors:ĊĊ246Ġ=Ġ2Ġ*Ġ
Rollout 91, Act: -2.254
ĠdeterminedĠupĠtoĠparametrization.ĠAssumingĠDĠisĠorigin,ĠinĠourĠcoordinateĠsystem.ĠHowever,ĠifĠwe
Rollout 61, Act: -2.304
*13)*Ġ(2*5).ĠSoĠprimeĠfactors:Ġ2^2Ġ*3Ġ*5
Rollout 62, Act: -2.369
13=>Ġyes.Ġ883ĠinĠprime?ĠYeah.ĊĊSoĠm=99:1
Rollout 62, Act: -2.392
517=11*47.ĠPrime.ĊĊk=3:Ġm=34.
Rollout 53, Act: -2.480
ĠtheĠcentroid?ĠMaybeĠcoordinatesĠsystemĠwhereĠcentroidĠisĠtheĠorigin,ĠorĠnot.ĠLet'sĠtryĠagain.ĊĊAlternative
Rollout 19, Act: -2.501
3ĠbutĠ>=6?Ġ83ĠitselfĠisĠprime,ĠsoĠonlyĠ1Ġand83.ĠSo
Rollout 20, Act: -2.559
Ġ2162?ĠLetĠmeĠcheck.ĊĊPrimeĠfactors:ĊĊ1806:Ġ2,
Rollout 76, Act: -2.762
=139.Ġ139ĠisĠprimeĠ(sinceĠit'sĠnotĠdivisibleĠbyĠ2,
Rollout 66, Act: -2.953
3ĠfactorsĠ13;Ġ13ĠisĠprime,ĠandĠ84ĠisĠ12*
Rollout 8, Act: -3.110
ĠtheyĠcanĠbeĠsimplified.ĠSinceĠ31ĠisĠprime.Ġ126ĠdividesĠbyĠ2,

Layer 25

GATE_PROJ

Top 16 Positive Activations
Rollout 97, Act: 2.812
ĠofĠN2O5}=ĠP0ĠVĠ/ĠRĠT}=ĠAssumingĠclosedĠvessel.ĊĊAtĠpressureĠP0
Rollout 75, Act: 2.674
Ġ32=Ġ2?ĠWaitĠhangĠon.ĠAreĠyouĠonlyĠevenĠsquaresĠeligible?ĊĊWait,Ġconsider
Rollout 97, Act: 2.447
ĠpartialĠpressureĠisĠproportionalĠtoĠconcentrationĠ(PVĠ=ĠnRT,ĠsoĠn/VĠ=ĠP/(RT)).ĠSo
Rollout 3, Act: 2.399
(3RT/M),ĠwhereĠRĠisĠtheĠgasĠconstant,ĠTĠisĠtheĠtemperatureĠinĠKelvin,ĠandĠM
Rollout 22, Act: 2.377
RT/(ÏĢM)),ĠwhereĠRĠisĠtheĠgasĠconstantĠandĠMĠisĠtheĠmolarĠmass.ĠWait,
Rollout 45, Act: 2.350
ĠthatĠdoesn'tĠmakeĠsense.ĠWait,ĠholdĠon.ĠLetĠmeĠcorrect.ĊĊWait,Ġwait,ĠrhsĠshould
Rollout 85, Act: 2.334
0)=4?ĠWait,ĠstartingĠtoĠgetĠconfused.ĊĊAlternatively,ĠtryĠtoĠdefineĠtheĠgridsĠasĠ2
Rollout 22, Act: 2.325
3ĠK):ĊĊv_meanĠ=Ġsqrt(8Ġ*ĠRĠ*ĠTĠ/Ġ(ÏĢĠ*ĠM))Ġ=Ġsqrt
Rollout 22, Act: 2.269
ĠbyĠsqrt(8RT/(ÏĢM)),ĠwhereĠRĠisĠtheĠgasĠconstantĠandĠMĠisĠtheĠmolar
Rollout 70, Act: 2.269
2ĠisĠinĠbase-7Ġdigits.ĠAre.ĊĊWaitĠN=315:6Ġ3Ġ
Rollout 10, Act: 2.266
Ġ1,ĠwhichĠisĠwinning.ĠHmm,Ġwait.ĠSoĠifĠfromĠn=5,ĠcanĠweĠmove
Rollout 55, Act: 2.266
ĠButĠactually,Ġno,ĠmatrixĠmultiplicationĠisĠmoreĠinvolved.ĠWait,Ġlet'sĠnotĠjumpĠtheĠgun.ĠLet
Rollout 22, Act: 2.247
3ĠK:ĊĊv_meanĠ=Ġsqrt(8Ġ*ĠRĠ*ĠTĠ/Ġ(ÏĢĠ*ĠM)),ĠwhereĊ
Rollout 22, Act: 2.246
8ĠK.ĊĊv_meanĠ=Ġsqrt(8Ġ*ĠRĠ*ĠTĠ/Ġ(ÏĢĠ*ĠM)).ĊĊR
Rollout 44, Act: 2.244
ĠpressureĠisĠnotĠtooĠfarĠfromĠtheĠinitial.ĠLetĠmeĠtakeĠp2=1.5Ġatm,ĠV
Rollout 10, Act: 2.240
ĠandĠodd.ĠButĠmaybeĠeveryĠotherĠpair.ĠLetĠmeĠthink.ĠAlternatively,ĠmaybeĠinĠeachĠtwoĠsteps,
Top 16 Negative Activations
Rollout 90, Act: -3.783
891928}ĊĊ**FinalĠAnswer**Ċ\boxed{1008919
Rollout 81, Act: -3.783
510/11}ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{510}{
Rollout 70, Act: -3.797
boxed{315}.ĊĊ**FinalĠAnswer**Ċ\boxed{315}ĊĊ<|im_start|>answerĊ
Rollout 38, Act: -3.799
Ġ\boxed{61}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{61}ĊĊ<|im_start|>answerĊAna
Rollout 93, Act: -3.806
Ġclue.ĠGivenĠtheĠprocessĠrigorouslyĠtheĠAnswerĠshouldĠbe:ĊĊ(1007Ġ/Ġ2^{
Rollout 53, Act: -3.810
{27}{2}}.ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{27}{2
Rollout 36, Act: -3.812
Ġ\boxed{192}ĊĊ**FinalĠAnswer**Ċ\boxed{192}ĊĊ<|im_start|>answerĊ
Rollout 11, Act: -3.826
(-1,Ġ3)}.ĊĊ**FinalĠAnswer**Ċ\boxed{(-1,Ġ3)}ĊĊ<|im_start|>
Rollout 40, Act: -3.826
Ġ\boxed{19}.ĊĊ**FinalĠAnswer**Ċ\boxed{19}ĊĊ<|im_start|>answerĊThe
Rollout 52, Act: -3.851
{-2013}.ĊĊ**FinalĠAnswer**Ċ\boxed{-2013}ĊĊ<|im_start|>answer
Rollout 71, Act: -3.854
Ġ\boxed{98}.ĊĊ**FinalĠAnswer**Ċ\boxed{98}ĊĊ<|im_start|>answerĊGiven
Rollout 2, Act: -3.871
Ġ\boxed{109}ĊĊ**FinalĠAnswer**Ċ\boxed{109}ĊĊ<|im_start|>answerĊ
Rollout 91, Act: -3.874
ĠisĠ\boxed{2}.ĊĊ**FinalĠAnswer**Ċ\boxed{2}ĊĊ<|im_start|>answerĊGivenĠtriangle
Rollout 97, Act: -3.889
{2.30}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{230}ĊĊ<|im_start|>answerĊ
Rollout 25, Act: -3.894
Ġ\text{Ġkg}}$ĊĊĊ**FinalĠAnswer**ĊTheĠasteroidĠwouldĠneedĠtoĠbeĠ$\boxed{4
Rollout 73, Act: -3.904
Ġ\boxed{398}ĊĊ**FinalĠAnswer**Ċ\boxed{398}ĊĊ<|im_start|>answerĊ

UP_PROJ

Top 16 Positive Activations
Rollout 33, Act: 2.421
Ġ+Ġ(-1)^mĠ|A_1ĠâĪ©ĠA_2ĠâĪ©Ġ...ĠâĪ©ĠA_
Rollout 79, Act: 2.404
*(Note:ĠWait,ĠmaybeĠ"ChangingĠHams"?ĠNotĠmakingĠsense.ĠWaitĠapologies,ĠafterĠtime-consuming
Rollout 64, Act: 2.390
ĠisĠtheĠdesiredĠanswer.ĊĊ**FinalĠAnswer**Ċ\boxed{36Ġ\left(2^{20
Rollout 23, Act: 2.371
ĠmightĠhaveĠanĠerror.ĊĊ**FinalĠAnswer**Ċ\boxed{B}ĊĊ<|im_start|>answerĊToĠdetermineĠtheĠactivity
Rollout 69, Act: 2.329
=204.ĊĊ**FinalĠAnswer**Ċ\boxed{204}ĊĊ<|im_start|>answerĊGivenĠthe
Rollout 72, Act: 2.315
ĠallĠevenĠpositiveĠintegers.ĊĊ**FinalĠAnswer**Ċ\boxed{\text{AllĠevenĠpositiveĠintegers}}ĊĊ<|im_start|>answer
Rollout 65, Act: 2.303
.ĠButĠsubtractingĊĊ**FinalĠAnswer**Ċ\boxed{40}ĊĊ<|im_start|>answerĊGivenĠaĠright
Rollout 76, Act: 2.296
3,ĠwaitĠ417:Ġ4+1+7=12,ĠsoĠdivisibleĠby
Rollout 72, Act: 2.292
Ġ1Ġ+Ġ2Re(a)=1ĠimpliesĠRe(a)=0,ĠwhichĠisĠnotĠpossibleĠforĠroots
Rollout 77, Act: 2.279
4,52)=Ġ[52/(52-14)]*266Ġ=Ġ(
Rollout 81, Act: 2.276
0/11}ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{510}{11
Rollout 72, Act: 2.262
Ġ1Ġ+Ġ2Re(a)=1ĠâĩĴĠRe(a)=0.ĠButĠRe(a)=0Ġimplies
Rollout 11, Act: 2.261
4.ĠButĠcĠ=Ġ-1/4ĠisĠreal,ĠwhichĠisĠprohibited.ĠNotĠallowed.ĠSo,
Rollout 84, Act: 2.259
5/14=Ġ(7/14Ġ-5/14)=2/14=
Rollout 14, Act: 2.256
ĠvectorsĠ/ĠweighetĊĊ**FinalĠAnswer**Ċ\boxed{Ġ\left(\int_0^\infty
Rollout 72, Act: 2.252
144°)=Ġcos(180°Ġ-36°)=Ġ-cos(36
Top 16 Negative Activations
Rollout 25, Act: -2.007
Ġa_mĠ=ĠG*(MĠ+Ġm)/r2.ĠHence,ĠinĠtheĠframeĠofĠreferenceĠofĠone
Rollout 17, Act: -2.011
Ġcos2θĠ-Ġμ_p]ĠdividedĠbyĠr3ĠmultipliedĠbyĠ|Ψ(r)|2,ĠthenĠintegrated
Rollout 97, Act: -2.015
ĠatmĠofĠN2O5.ĠForĠeveryĠ2ĠunitsĠofĠpressureĠ(atm)ĠofĠN2
Rollout 9, Act: -2.017
ĠbetterĠway.ĠSinceĠeachĠsideĠisĠ10Ġunits.ĠSoĠAIĠwouldĠbeĠvertical,ĠIMĠhorizontal,
Rollout 96, Act: -2.026
ĠpassesĠthroughĠpointĠA(24,Ġ-1).ĠLet'sĠconfirmĠtheĠequationĠforĠlineĠL.ĠThe
Rollout 46, Act: -2.030
ĠtwoĠtermsĠ-GĠM1Ġ/ĠÏģ12Ġ*ĠâĪĤÏģ1/âĪĤxĠisĠthe
Rollout 25, Act: -2.066
Ġmasses,ĠFĠ=ĠG*M*mĠ/Ġd^2.ĠThen,ĠaccelerationĠonĠtheĠasteroidĠisĠF/m
Rollout 87, Act: -2.127
ĠisĠsqrt(x2Ġ+Ġy2Ġ+Ġz2).ĠTherefore,ĠtheĠintegralĠbecomesĠaĠtripleĠintegralĠoverĠx
Rollout 25, Act: -2.156
ĠasteroidĠisĠFĠ=ĠG*M*mĠ/Ġd^2,ĠwhereĠMĠisĠEarth'sĠmass,ĠmĠis
Rollout 81, Act: -2.166
:ĊĊAB=Ġsqrt{32Ġ+h2Ġ}ĊĊAC=Ġsqrt{172Ġ+h2
Rollout 25, Act: -2.194
ĠaccelerationĠdueĠtoĠgravityĠisĠG*MĠ/Ġd^2,ĠsolelyĠdependentĠonĠEarth'sĠmass.ĊĊWait,
Rollout 38, Act: -2.196
Ġ+ĠW.ĊĊButĠwait,ĠAnaĠhasĠspeedĠ8.6Ġm/s,ĠsoĠtheĠtimeĠwouldĠbe
Rollout 25, Act: -2.225
ĠbeĠacceleratingĠwithĠG*(MĠ+Ġm)/r2.ĊĊTherefore,ĠtheĠrelativeĠmotionĠcanĠbeĠmodeledĠusing
Rollout 25, Act: -2.227
ĠrelativeĠaccelerationĠisĠG*(MĠ+Ġm)/d2.ĠTherefore,ĠinĠtheĠtwo-bodyĠproblem,Ġthe
Rollout 97, Act: -2.227
ĠrateĠofĠtheĠreactionĠisĠgivenĠwithĠrateĠconstantĠ5ĠxĠ10^-4Ġs^-1.
Rollout 25, Act: -2.314
ĠasteroidĠwouldĠbeĠaĠ=ĠG*mĠ/Ġd^2,ĠwhereĠdĠisĠtheĠdistanceĠbetweenĠthem.ĠLet

DOWN_PROJ

Top 16 Positive Activations
Rollout 10, Act: 2.289
ĠcanĠhaveĠsubtractionsĠonlyĠupĠtoĠsĠ<=n.ĠForĠeachĠn,Ġit'sĠaĠfiniteĠnumber
Rollout 90, Act: 2.279
Ġk=2Ġdigits.ĠWaitĠno:ĠholdĠon.ĠWait,ĠforĠaĠ5-digitĠnumber:ĠD
Rollout 79, Act: 2.229
"ĠisĠaĠcontainer,ĠbutĠI'mĠnotĠsure.ĊĊWell,Ġ"minorĠchangesĠin"ĠwouldĠbeĠM
Rollout 90, Act: 2.216
2Ġdigits.ĠWaitĠno:ĠholdĠon.ĠWait,ĠforĠaĠ5-digitĠnumber:ĠD1ĠD
Rollout 3, Act: 2.202
,ĠtemperatureĠmustĠbeĠinĠKelvinĠinĠtheĠformulae.ĊĊHence,ĠgivenĠtheĠconfusion,ĠperhapsĠtheĠproblem
Rollout 8, Act: 2.201
ĠWait,ĠholdĠon.ĠWait,Ġwait,Ġno.ĠIfĠsheĠisĠlayingĠoutĠ5ĠredĠcards,
Rollout 8, Act: 2.200
0.ĠWait,ĠholdĠon.ĠWait,Ġwait,Ġno.ĠIfĠsheĠisĠlayingĠoutĠ5Ġred
Rollout 33, Act: 2.195
\),ĠatĠleastĠoneĠofĠtheĠfirstĠ\(k\)ĠtermsĠofĠtheĠpermutationĠisĠgreaterĠthanĠ\(k\
Rollout 0, Act: 2.192
ĠtoĠtheirĠexponents)..ĠWait,ĠconfusionĠarisesĠhere.ĊĊWait,Ġactually,ĠweĠneedĠtoĠbeĠprecise.
Rollout 44, Act: 2.184
ĠprocessĠtoĠbeĠbothĠadiabaticĠandĠisothermal,ĠQ=0ĠandĠÎĶU=0Ġ(
Rollout 10, Act: 2.140
Ġs)Ġ|ĠsĠisĠaĠsquare,Ġs<=n}.ĊĊSoĠlet'sĠcomputeĠtheĠGrundyĠnumbers
Rollout 20, Act: 2.136
Ġ1.ĠWait,ĠnoâĢĶWait,Ġwait.ĊĊWait,Ġno.ĠLet'sĠsupposeĠthatĠp^n
Rollout 8, Act: 2.125
Ġ10.ĠWait,ĠholdĠon.ĠWait,Ġwait,Ġno.ĠIfĠsheĠisĠlayingĠoutĠ
Rollout 44, Act: 2.110
ĠtoĠV2.ĊĊWorkĠdoneĠbyĠgasĠinĠisothermalĠreversibleĠexpansion:ĠW_isoĠ=ĠnRTĠln(V
Rollout 44, Act: 2.107
ĠthanĠirreversible.ĊĊSoĠinĠthatĠcase,ĠsinceĠtheĠreversibleĠworkĠisĠlarger,ĠthenĠ[A]ĠwouldĠbe
Rollout 26, Act: 2.094
ĠsoĠthatâĢĻsĠwhyĠT=298ĠK.ĊĊThus,ĠinĠtheĠcalculationĠforĠÎĶH,Ġthe
Top 16 Negative Activations
Rollout 17, Act: -2.451
ÏĢĠa03Ġ)ĊĊButĠμ0ĠhasĠunitsĠT·m/A,Ġμ_pĠhasĠunitsĠJ/T
Rollout 59, Act: -2.465
16]Ġ/Ġ282ĊĊComputeĠnumerator:ĊĊ9Ġ+Ġ100Ġ=Ġ1
Rollout 17, Act: -2.477
/AĠ*ĠJ/TĠ)Ġ/Ġm3Ġ=Ġ(J·m)Ġ/Ġ(A·m3)
Rollout 96, Act: -2.486
Ġ/Ġ(169*13)ĊĊComputeĠnumerator:ĊĊ-4106*12=
Rollout 17, Act: -2.571
Ġ)Ġ=Ġkg/(A·s2)Ġ=ĠT,ĠsinceĠ1ĠTĠ=Ġkg/(A·
Rollout 97, Act: -2.582
Ġ2y)ĊĊButĠsolvingĠthis,ĠweĠcanĠrearrangeĠvariables:ĊĊdy/(1Ġ-2y)
Rollout 3, Act: -2.593
8*R*T)/(ÏĢ*M)),ĠRĠisĠinĠJ/(mol·K),ĠwhichĠisĠsameĠasĠ(
Rollout 17, Act: -2.597
Ġ*ĠJ/TĠ?ĊĊWait,Ġμ0ĠhasĠunitsĠT·m/A,Ġμ_pĠhasĠunitsĠJ/T
Rollout 97, Act: -2.666
ĠrearrangeĠvariables:ĊĊdy/(1Ġ-2y)Ġ=Ġ(kĠ/Ġ2)ĠdtĊĊInt
Rollout 97, Act: -2.682
ĠsolvingĠthis,ĠweĠcanĠrearrangeĠvariables:ĊĊdy/(1Ġ-2y)Ġ=Ġ(kĠ/Ġ
Rollout 80, Act: -2.696
So,ĠtheĠvolumeĠofĠtheĠinnerĠcoreĠisĠ(4/3)ÏĢr3.ĠLet'sĠcalculate
Rollout 17, Act: -2.758
Ġ)Ġ=Ġkg/(A·s2)Ġ=ĠT,ĠwhichĠmatches.ĠCorrect.ĠSoĠtheĠresultĠis
Rollout 80, Act: -2.872
ĠtheĠvolumeĠofĠtheĠinnerĠcoreĠisĠ(4/3)ÏĢr3.ĠLet'sĠcalculateĠthat:ĊĊ
Rollout 17, Act: -2.889
·m/AĠ*ĠJ/TĠĊĊButĠJĠ=ĠN·mĠ=Ġ(kg·m2/s2
Rollout 76, Act: -2.903
ĠentirelyĠredĠsquares.ĠItĠmightĠbeĠeasierĠtoĠcomputeĠtheĠnumberĠofĠcoloringsĠthatĠhaveĠatĠleastĠoneĠ
Rollout 97, Act: -3.099
angeĠvariables:ĊĊdy/(1Ġ-2y)Ġ=Ġ(kĠ/Ġ2)ĠdtĊĊIntegrate

Layer 26

GATE_PROJ

Top 16 Positive Activations
Rollout 20, Act: 3.878
'sĠsee,Ġ43*900=38,700;Ġ43
Rollout 96, Act: 3.871
1ĠCheck:ĊĊ13*8000=104,000ĊĊ13
Rollout 96, Act: 3.809
3:Ġ13*6000=78,000,Ġ88
Rollout 81, Act: 3.793
*561:ĊĊ-ĠCalculateĠ40*561=22,440
Rollout 96, Act: 3.755
13:Ġ13*649=843,ĠandĠmyĠpreviousĠstepĠhadĠ
Rollout 53, Act: 3.723
146:ĊĊ146*13=1898ĠâĨĴĠ1971
Rollout 96, Act: 3.717
Ġcompute:ĊĊ4106*100=410600ĊĊ410
Rollout 36, Act: 3.673
624:ĊĊ30,000*30,000=900
Rollout 20, Act: 3.643
:ĊĊFirstĠcomputeĠ40,000*47=1,880,0
Rollout 36, Act: 3.636
Ġlet'sĠcheck:ĊĊ10,000*10,000=100
Rollout 90, Act: 3.608
:Ġ2,000*44=88,000;Ġ18
Rollout 17, Act: 3.590
400*3.1416=1256.64ĊĊ44
Rollout 96, Act: 3.565
37ĠdivisibleĠby13:Ġ13*649=843,ĠandĠmy
Rollout 36, Act: 3.557
064:Ġ8,000*8,000Ġ=Ġ64,
Rollout 96, Act: 3.553
.ĠWaitĠ13*2000=26,000,ĠremainingĠ2
Rollout 62, Act: 3.553
?ĠCalculatingĠ23*100=2300,Ġ23*7
Top 16 Negative Activations
Rollout 28, Act: -3.215
ĠtoĠnon-degenerateĠhyperbolas.ĠIfĠso,ĠthenĠtheĠstatementĠwouldĠrequireĠthatĠallĠnon-degenerate
Rollout 43, Act: -3.223
ĠisĠfirst,ĠthenĠtargetĠqubit.ĊĊInĠthatĠcase,ĠtheĠinputĠisĠ(controlĠqubit)âĬĹ
Rollout 11, Act: -3.226
ĠtoĠbothĠpĠandĠqĠbeingĠreal.ĊĊHence,ĠmaybeĠsuchĠaĠcaseĠisĠimpossible,ĠsoĠperhapsĠwe
Rollout 66, Act: -3.227
ĠsolutionĠexistsĠunderĠthisĠapproach,ĠbutĠtheĠproblemĠstatesĠthatĠsuchĠaĠtriangleĠexists.ĊĊWait,ĠperhapsĠIĠmade
Rollout 38, Act: -3.230
Ġcan'tĠbeĠright.ĊĊHence,Ġthere'sĠaĠflawĠinĠtheĠassumptionĠthatĠtheĠpathĠforĠAnaĠis
Rollout 35, Act: -3.246
8.ĠĊĊButĠwait:ĠButĠtheĠproblemĠstatesĠthatĠtheĠequationĠP(n)=nĠ+3ĠhasĠtwo
Rollout 29, Act: -3.276
ĠmilkĠandĠhandle'sĠmilkĠareĠseparate.ĊĊInĠthatĠcase,ĠtheĠpressureĠatĠtheĠbottomĠofĠtheĠmainĠjug
Rollout 66, Act: -3.282
ĠbeĠbothĠodd.ĠTherefore,ĠtheĠerrorĠarisesĠhere.ĠTherefore,ĠmĠandĠnĠmustĠhaveĠoppositeĠparity.
Rollout 0, Act: -3.284
,ĠnumeratorĠaĠandĠdenominatorĠbĠmustĠsatisfyĠthat.ĠHowever,ĠaĠdifferentĠwayĠtoĠphraseĠthisĠis:Ġa
Rollout 50, Act: -3.304
)\)ĠisĠindeedĠtheĠcorrectĠvalue.ĠThus,ĠtheĠfinalĠanswerĠisĠ\(11Ġ+Ġ10
Rollout 64, Act: -3.304
,ĠbutĠtheĠproblemĠstatesĠthatĠtheĠsequenceĠisĠofĠpositiveĠintegers.ĠSoĠallĠtermsĠa_nĠmustĠbeĠpositive
Rollout 66, Act: -3.312
Ġâīł1.ĠThus,Ġproblem.ĊĊHence,ĠtheĠapproachĠisĠleadingĠtoĠnonĠprimitiveĠtriples.ĠWhichĠsuggests
Rollout 20, Act: -3.353
ĠanyĠpositiveĠintegerĠa,ĠincludingĠnon-primes.ĠHowever,ĠsinceĠÏĥĠisĠmultiplicative,ĠwhichĠallowsĠus
Rollout 95, Act: -3.365
ĠbeingĠtrapezoidsĠexceptĠtheĠfirstĠtriangle.ĠTherefore,ĠifĠeachĠtrapezoidĠhasĠtheĠsame
Rollout 31, Act: -3.376
Ġm.ĠWaitĠaĠminute,ĠdidĠweĠmaybeĠoverlookĠtheĠroleĠofĠm?ĊĊHmm,ĠthatĠmayĠbeĠa
Rollout 84, Act: -3.440
.5,ĠsoĠthat'sĠimpossible.ĠThus,ĠtheĠonlyĠpossibleĠvalidĠinterpretationĠisĠtheĠfirstĠone.ĠTherefore

UP_PROJ

Top 16 Positive Activations
Rollout 84, Act: 1.649
2}Ġ-Ġ2Ġ\timesĠ\frac{m}{n}Ġ\).ĊĊAccordingĠtoĠtheĠproblem,
Rollout 83, Act: 1.552
ĠtheĠnumberĠofĠdistinctĠarrangementsĠisĠ1296Ġ/Ġ9Ġ=144.ĊĊButĠlet
Rollout 83, Act: 1.529
}{9}Ġ\left(Ġ1296Ġ+Ġ0Ġ+Ġ0Ġ+Ġ\cdots
Rollout 36, Act: 1.518
Ġ$BP=60\sqrt{10}$Ġ,Ġ$CP=60\sqrt{5
Rollout 36, Act: 1.505
sqrt{10}$Ġ,Ġ$CP=60\sqrt{5}$Ġ,Ġ$DP=1
Rollout 80, Act: 1.471
1}Ġ\,Ġ\text{neutrinos/s/cm}^2Ġ\timesĠ4.5
Rollout 79, Act: 1.470
Ġit'sĠphrasedĠisĠ"NoticedĠminorĠchangesĠinĠinvestigators'Ġfacility,"ĠmakingĠ"minorĠchangesĠin
Rollout 83, Act: 1.445
.ĠHence,ĠnumberĠofĠorbitsĠis1296/9=144.ĊĊThus,Ġanswering
Rollout 9, Act: 1.431
ĠtriangleĠGEMĠandĠsquareĠAIMEĠisĠ80.ĠSinceĠtheĠsquareĠisĠfromĠ(0,0
Rollout 27, Act: 1.431
54ĠÃĹĠ10^-17ĠJ,ĠcalculateĠtheĠrmsĠspeedĠofĠtheĠBeĠnuclei,ĠV
Rollout 5, Act: 1.423
Ġ(HĠ-Ġh)Ġ=Ġ62.5H.ĊĊSoĠweĠhaveĠtwoĠequations:ĊĊ1.
Rollout 66, Act: 1.420
ĠsumĠofĠthisĠseriesĠisĠgivenĠasĠ\(6p\).ĠTheĠkeyĠequationĠderivedĠfromĠtheĠgeometricĠseriesĠsum
Rollout 95, Act: 1.412
ĠABCĠ[Ġ(k/2450)^2Ġ-Ġ(Ġ(k-1)/245
Rollout 5, Act: 1.407
Ġx)/2Ġ*ĠhĠ=Ġ62.5H.ĠTheĠremainingĠtrapezoidĠhasĠbases
Rollout 9, Act: 1.407
\)ĠandĠsquareĠ\(AIME\)ĠisĠ80ĠsquareĠunits.ĠWeĠneedĠtoĠfindĠtheĠlengthĠof
Rollout 53, Act: 1.403
:ĊĠĠĠ-ĠUsingĠ\(DEĠ=Ġ2.5\),ĠweĠgetĠ\((aĠ+Ġc)^
Top 16 Negative Activations
Rollout 74, Act: -2.986
,ĠandĠ$\theta$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠderiveĠanĠexpression
Rollout 92, Act: -2.992
ĠpowerĠofĠhisĠglassesĠlenses?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠthisĠout
Rollout 22, Act: -2.997
2)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠtoĠcalculateĠtheĠmean
Rollout 46, Act: -2.999
^{2}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠtoĠfindĠtheĠsecond
Rollout 31, Act: -3.007
2:Ġx^3Ġ+Ġx^2.ĠSubtractĠfromĠdividend:ĊĊ(x^3Ġ-0x^
Rollout 3, Act: -3.011
n)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculateĠtheĠmeanĠmolecular
Rollout 17, Act: -3.014
$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackleĠthisĠproblem
Rollout 97, Act: -3.016
Ġ$\mathrm{Y}$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠhaveĠthis
Rollout 62, Act: -3.022
.ĠSolve.ĊĊTryĠmĠvalues:ĊĊm=0:-2âī¡-2âīł0ĊĊm=1
Rollout 29, Act: -3.024
Ġ\rhoĠgH$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 25, Act: -3.025
ĠisĠtheĠimpactĠparameter.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠthisĠproblemĠout
Rollout 47, Act: -3.069
ĠvalueĠofĠ$N$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠthere'sĠthisĠproblemĠaboutĠa
Rollout 26, Act: -3.079
mol}^{-1}$Ġ]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthisĠproblem
Rollout 80, Act: -3.110
ĠvalueĠofĠ$N$Ġ?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthis
Rollout 25, Act: -3.133
$ĠisĠtheĠimpactĠparameter.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠthisĠproblem
Rollout 27, Act: -3.167
ĠalongĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthisĠproblemĠhere

DOWN_PROJ

Top 16 Positive Activations
Rollout 22, Act: 2.819
333e3Ġ*1e3âīĪĠ(19.820576Ġ/
Rollout 22, Act: 2.669
6.333e-3ĠâīĪĠ(18,168.776
Rollout 54, Act: 2.633
ophilesĠgivenĠare:ĊĊ1.Ġ4-methylcyclohexan-1-olateĊ
Rollout 4, Act: 2.630
990Ġ*Ġ2985=Ġ(2000Ġ-10)*(30
Rollout 54, Act: 2.530
ButĠwithĠstericĠhindrance:Ġ4-methylcyclohexan-1-olateĠis
Rollout 22, Act: 2.530
ĠdoĠ66Ġ*Ġ273Ġ=Ġ(60*273)Ġ+Ġ(6
Rollout 36, Act: 2.518
?ĊĊ19,360^2=Ġ(19,000Ġ+Ġ36
Rollout 38, Act: 2.496
,Ġ196/672=Ġ(28*7)/(28*24
Rollout 13, Act: 2.487
09ĊĊ1.03^4Ġ=Ġ(1.0609)^2ĊĊCompute
Rollout 70, Act: 2.478
2*51=102.ĠCorrect,ĠwhichĠisĠtheĠexampleĠgiven.ĊĊd1=1
Rollout 48, Act: 2.447
00Ġ+Ġ0xB612Ġ=Ġ(0x4500Ġ+Ġ0xB
Rollout 22, Act: 2.416
0.006333ĊĊ=Ġ(18,157.776
Rollout 54, Act: 2.375
ĠthanĠwater.ĠWait,ĠbutĠcomparingĠ4-methylcyclohexanol'sĠpKaâĢĶprobably
Rollout 3, Act: 2.372
3Ġ*Ġ66.512Ġ=Ġ(200Ġ+70Ġ+3)*6
Rollout 39, Act: 2.349
ĠisĠ1ĠfollowedĠbyĠ322ĠzerosĠminusĠ10.ĠSoĠtheĠresultĠwillĠbeĠa
Rollout 69, Act: 2.317
3Ġ-3Ġ)/12.ĠThat'sĠcorrect.ĊĊTherefore,ĠS=Ġ(2âĪļ3Ġ-
Top 16 Negative Activations
Rollout 60, Act: -1.967
ĠtheĠsoundĠatĠtheĠfrequencyĠitĠhasĠreceived).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠto
Rollout 38, Act: -1.973
ĠsidesĠareĠtheĠnorth,ĠsouthĠedges;ĠtheĠshorterĠsidesĠareĠeastĠandĠwestĠedges.ĠWait,ĠthatĠseems
Rollout 76, Act: -1.993
What'sĠtheĠintersectionĠofĠS1,ĠS2,ĠandĠS3?ĠEachĠpairĠofĠtheseĠoverlaps:ĊĊ
Rollout 76, Act: -2.027
ĠtwoĠ2x2ĠsquaresĠareĠbothĠallĠred.ĠHowĠmanyĠsuchĠpairsĠareĠthere,ĠandĠwhat's
Rollout 15, Act: -2.030
i-1,t}Ġ+Ġx_{i+1,t}ĠmodĠ2ĊĊThisĠisĠsimilarĠto
Rollout 16, Act: -2.047
8).Ġ1953=68*28Ġ+Ġ49ĊĊ68=
Rollout 42, Act: -2.065
utylĠorĠcyclopropylĠside?ĠOrĠwait,ĠmaybeĠtheĠadjacentĠcarbonsĠareĠnotĠonĠthe
Rollout 87, Act: -2.072
0}$,ĠandĠanyĠnecessaryĠnumericalĠconstants.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠto
Rollout 33, Act: -2.113
,ĠaccordingĠtoĠformulaĠforĠn=4:ĊĊUsingĠinclusion-exclusion:ĊĊ24Ġ-16(Ġm
Rollout 22, Act: -2.142
ĠhydrogenĠ(H2)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠto
Rollout 83, Act: -2.148
ĠofĠresiduesĠ0,1,2ĠtoĠGroup0,1,2),ĠtheĠnumberĠofĠdistinctĠlabel
Rollout 76, Act: -2.177
ĠareĠbothĠallĠred.ĠHowĠmanyĠsuchĠpairsĠareĠthere,ĠandĠwhat'sĠtheĠnumberĠofĠcoloringsĠfor
Rollout 74, Act: -2.182
_{s}$,ĠandĠ$\theta$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠto
Rollout 76, Act: -2.188
ĠusingĠtheĠinclusion-exclusionĠprinciple.ĊĊSo,ĠapplyĠinclusion-exclusion.ĠLetĠtheĠfourĠ2x2
Rollout 3, Act: -2.205
onĠ(Rn)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculate
Rollout 76, Act: -2.268
ĠweĠneedĠtoĠadjustĠforĠovercountingĠusingĠtheĠinclusion-exclusionĠprinciple.ĊĊSo,ĠapplyĠinclusion-exclusion

Layer 27

GATE_PROJ

Top 16 Positive Activations
Rollout 97, Act: 2.872
?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠhaveĠthisĠdecompositionĠreaction:Ġ2
Rollout 23, Act: 2.839
Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠstepĠbyĠstep.ĠSo,Ġwe
Rollout 26, Act: 2.819
Ġ]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthisĠproblemĠout.ĠSo,Ġwe
Rollout 91, Act: 2.794
Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠgeometryĠproblemĠstepĠbyĠstep.ĠTheĠquestion
Rollout 5, Act: 2.782
Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠneedĠtoĠsolveĠthisĠproblemĠaboutĠa
Rollout 65, Act: 2.760
?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee,ĠI'veĠgotĠthisĠproblemĠhereĠaboutĠa
Rollout 64, Act: 2.738
Ġ?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠoutĠthisĠproblem.ĠWeĠhaveĠa
Rollout 54, Act: 2.718
Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestionĠstepĠbyĠstep.ĠFirst,ĠI
Rollout 52, Act: 2.695
?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠaboutĠyummyĠintegers.ĠTheĠquestionĠis
Rollout 8, Act: 2.689
Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠWeĠneedĠtoĠfindĠtheĠprobabilityĠthatĠwhen
Rollout 58, Act: 2.669
*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠSoĠPatchouliĠhasĠanĠexamĠwith
Rollout 67, Act: 2.636
.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠTheĠproblemĠsaysĠthatĠtheĠsequenceĠstartsĠwith
Rollout 40, Act: 2.623
Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblemĠstepĠbyĠstep.ĠSo,Ġwe
Rollout 62, Act: 2.613
.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblem:ĠFindĠtheĠleastĠpositiveĠintegerĠ\(
Rollout 95, Act: 2.582
.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠneedĠtoĠsolveĠthisĠgeometryĠproblemĠabout
Rollout 31, Act: 2.570
.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠtackleĠthisĠproblem.ĠSo,ĠweĠhave
Top 16 Negative Activations
Rollout 93, Act: -3.484
6)/16=2*8/16=1).ĠForĠm=3:3*(
Rollout 0, Act: -3.493
128.ĠAlternatively,ĠtotalĠnumberĠofĠsubsetsĠisĠ2^8=256,Ġeach
Rollout 93, Act: -3.512
ĠformulaĠgivesĠsqrt(3/3.14âīĪ0.97ĠwhereasĠactualĠformulaĠgives33
Rollout 64, Act: -3.518
Ġthis.ĊĊButĠ36=4*9,ĠsoĠmayĠbeĠwrittenĠas9*4*(2
Rollout 69, Act: -3.521
Alternatively,Ġ4Ġ-ĠâĪļ3ĠâīĪ4Ġ-1.732âīĪ2.2
Rollout 39, Act: -3.521
9,ĠtotalĠisĠ318*1Ġ+789,ĠbutĠhowĠexactly?ĊĊAlternatively,
Rollout 73, Act: -3.529
}Ġ+2=9*2048Ġ+2=18432Ġ+2=
Rollout 86, Act: -3.547
/8.ĠNumericalĠvalue:Ġpi^3âīĪ31.006,ĠsoĠ3
Rollout 33, Act: -3.568
.ĊĊButĠwaitĠ6!Ġequals720Ġ,ĠtheĠanswerĠisĠ461ĠwhichĠmakesĠsense
Rollout 0, Act: -3.611
4,ĠwhichĠisĠ2^{3-1}=4.ĊĊSimilarly,ĠifĠnĠisĠaĠsquare-free
Rollout 70, Act: -3.639
?ĠWait,ĠwaitĠN=35*9Ġ+16=315Ġ+16=
Rollout 81, Act: -3.653
39270=ĠdivideĠnumeratorĠandĠdenominatorĠbyĠ10:Ġ12342
Rollout 51, Act: -3.712
2/7=Ġ(14Ġ-18)/63Ġ=Ġ(-4)/63.ĊĊTherefore
Rollout 0, Act: -3.745
ĠcountĠabove).ĠButĠ2^{3-1}=Ġ4Ġcorrect.ĠHence,ĠTherefore,ĠtheĠformula
Rollout 75, Act: -3.811
ĠSoĠthenĠscalingĠback,Ġsqrt(4ÃĹ4)=4*Ġ2^2.ĠThisĠisĠsimilar
Rollout 86, Act: -3.812
Ġpi^3âīĪ31.006,ĠsoĠ31.006/4

UP_PROJ

Top 16 Positive Activations
Rollout 2, Act: 2.337
8,ĠsoĠ10^kĠmodĠ8ĠisĠ2^kĠmodĠ8.ĊĊThus
Rollout 19, Act: 2.292
ButĠalso,ĠnoteĠthatĠcongruenceĠmodulusĠk.ĠSoĠneedĠx2Ġâī¡17Ġmodk
Rollout 19, Act: 2.252
.ĊĊButĠalso,ĠnoteĠthatĠcongruenceĠmodulusĠk.ĠSoĠneedĠx2Ġâī¡17Ġmod
Rollout 2, Act: 2.243
Ġ(10^kĠ-1)ĠmodĠ8ĠisĠ(2^kĠ-1)ĠmodĠ
Rollout 2, Act: 2.229
SinceĠ10Ġâī¡Ġ2ĠmodĠ8,ĠsoĠ10^kĠmodĠ8
Rollout 2, Act: 2.216
^kĠ-1)Ġâī¡Ġ-1Ġmod8.ĊĊTherefore,ĠforĠk=1:Ġ(2
Rollout 81, Act: 2.201
LetĠmeĠcalculateĠangleĠBACĠbyĠvectors.ĠABĠvectorĠfromĠAĠtoĠB:Ġ(-3,Ġ-h
Rollout 71, Act: 2.185
Ġ+Ġm,ĠwhereĠm=Ġ-abc.ĠOkay.ĊĊWeĠneedĠintegersĠa,Ġb,ĠcĠwithĠa
Rollout 73, Act: 2.181
4,8,...,1024}.ĠSoĠforĠeachĠpairĠwithĠlargerĠelementĠ2^j
Rollout 21, Act: 2.144
?ĊĊWait,ĠsupposeĠf:ĠRâĨĴZ.ĠThen,Ġf(f(x)Ġ+Ġy)ĠisĠequal
Rollout 81, Act: 2.140
ACĠbyĠvectors.ĠABĠvectorĠfromĠAĠtoĠB:Ġ(-3,Ġ-h).ĠACĠvectorĠfromĠA
Rollout 2, Act: 2.140
Ġ8.ĊĊComputeĠ2^kĠmodĠ8:ĊĊNoticeĠthatĠ2^1Ġ=Ġ2
Rollout 71, Act: 2.127
ĠLetĠmeĠcheck.ĊĊSupposeĠcĠisĠodd.ĠSoĠweĠhadĠbefore,ĠafterĠsubstitution,ĠtheĠequationĠ
Rollout 68, Act: 2.107
ĠkĠnegative,ĠxĠnegative:ĠthusĠkxĠpositive).ĠButĠourĠdomainĠwasĠindeedĠxĠinĠ(-2,
Rollout 35, Act: 2.106
0Ġ+19Ġ+3=22.ĠWhichĠalignsĠwithĠnĠ+3.ĊĊSimilarly,Ġat
Rollout 19, Act: 2.098
Ġ+Ġ20)/2,ĠthenĠmoduloĠk,ĠweĠcanĠcomputeĠk^2Ġmodk=0
Top 16 Negative Activations
Rollout 27, Act: -1.613
ĠSolvingĠforĠtheĠone-dimensionalĠrmsĠvelocity:ĊĠĠĠĠĠ\[ĊĠĠĠĠĠv_{z,Ġ\text{rms
Rollout 83, Act: -1.617
ĠsumĠofĠeveryĠthreeĠconsecutiveĠverticesĠisĠaĠmultipleĠofĠ3,ĠweĠstartĠbyĠanalyzingĠtheĠresiduesĠmoduloĠ
Rollout 83, Act: -1.617
ĠofĠaĠregularĠnonagonĠwithĠtheĠdigitsĠ1ĠthroughĠ9ĠsuchĠthatĠtheĠsumĠofĠeveryĠthreeĠconsecutive
Rollout 2, Act: -1.623
,ĠthenĠcombineĠtheĠresultsĠusingĠtheĠChineseĠRemainderĠTheorem.ĊĊ1.Ġ**ModuloĠ8ĠCalculation
Rollout 17, Act: -1.623
pi}Ġ\left(Ġ\frac{3(\mathbf{\mu}_pĠ\cdotĠ\mathbf
Rollout 92, Act: -1.626
ĠToĠfindĠtheĠrefractiveĠpowerĠofĠtheĠlenses,ĠweĠuseĠtheĠlensĠformulaĠ\(Ġ\frac{1}{
Rollout 40, Act: -1.632
ĠconsistsĠofĠaĠcounterclockwiseĠrotationĠbyĠ\(\pi/4\)ĠradiansĠaboutĠtheĠoriginĠfollowedĠby
Rollout 80, Act: -1.678
,Ġ\text{cm}ĊĠĠĠ\]ĊĠĠĠ\[ĊĠĠĠ\text{Volume}Ġ=Ġ\frac
Rollout 29, Act: -1.678
ĠatĠrest,ĠtheĠpressureĠatĠaĠdepthĠ\(ĠhĠ\)ĠisĠgivenĠbyĠ\(ĠPĠ=Ġ\rhoĠgh
Rollout 20, Act: -1.694
Ġ\)ĠmustĠbeĠtheĠleastĠcommonĠmultipleĠ(LCM)ĠofĠ43,Ġ42,
Rollout 20, Act: -1.703
,Ġ\(ĠnĠ\)ĠmustĠbeĠtheĠleastĠcommonĠmultipleĠ(LCM)ĠofĠ43,Ġ
Rollout 16, Act: -1.719
omialĠtheoremĠandĠswappingĠtheĠorderĠofĠsummation,ĠweĠget:ĊĊ\[ĊE[T]Ġ=Ġ\sum
Rollout 92, Act: -1.729
0.615}ĊĠĠĠ\]ĊĠĠĠ\[ĊĠĠĠ\frac{1}{f}Ġ=
Rollout 20, Act: -1.734
)ĠmustĠbeĠtheĠleastĠcommonĠmultipleĠ(LCM)ĠofĠ43,Ġ42,Ġ
Rollout 26, Act: -1.803
4.8Ġ\,Ġ\text{K}Ġ=Ġ296.0Ġ\,Ġ\text
Rollout 29, Act: -1.882
Ġ=Ġ\rhoĠghĠ\),ĠwhereĠ\(Ġ\rhoĠ\)ĠisĠtheĠfluidĠdensity,Ġ\(ĠgĠ\)

DOWN_PROJ

Top 16 Positive Activations
Rollout 20, Act: 3.054
43ĠandĠ47,ĠthenĠtakeĠLCM.ĠLetĠmeĠfirstĠworkĠmoduloĠ43.ĊĊ
Rollout 20, Act: 2.678
ĠintoĠ2Ġ*Ġ23.ĊĊSoĠLCM(42,46)Ġ=Ġ2
Rollout 20, Act: 2.662
ĠworksĠbothĠmod43Ġand47ĠisĠLCM(1806,2162
Rollout 74, Act: 2.555
.ĠLetĠmeĠdefineĠtheĠresultantĠfieldĠasĠtheĠvectorĠsumĠofĠH_eĠandĠtheĠship'sĠfield.ĊĊLet
Rollout 20, Act: 2.498
62.ĠSo,ĠnĠmustĠbeĠtheĠleastĠcommonĠmultipleĠofĠ1806ĠandĠ2
Rollout 20, Act: 2.488
,46Ġmod47).ĠTherefore,ĠLCM(47,46)=216
Rollout 20, Act: 2.456
.ĊĊTherefore,ĠtheĠminimalĠsuchĠnĠisĠtheĠleastĠcommonĠmultipleĠofĠtheĠminimalĠnĠthatĠworksĠmodĠ4
Rollout 20, Act: 2.455
2Ġand43.ĊĊSo,nĠshouldĠbeĠLCM(42,43)=180
Rollout 20, Act: 2.452
Thus,Ġ\(ĠnĠ\)ĠmustĠbeĠtheĠleastĠcommonĠmultipleĠ(LCM)ĠofĠ43,
Rollout 20, Act: 2.428
Ġp,ĠnĠmustĠbeĠaĠmultipleĠofĠtheĠleastĠcommonĠmultipleĠofĠtheĠmultiplicativeĠordersĠofĠallĠpĠmodulo
Rollout 20, Act: 2.426
47ĠandĠmod46.Ġn=LCM(46,47)=216
Rollout 20, Act: 2.411
2162.ĊĊTotalĠminimalĠnĠisĠLCM(1806,2162
Rollout 20, Act: 2.400
.ĠTherefore,ĠtheĠminimalĠsuchĠnĠisĠtheĠleastĠcommonĠmultipleĠofĠ42ĠandĠ46.
Rollout 20, Act: 2.379
Ġconditions,ĠnĠmustĠbeĠâī¡0ĠmodĠLCM(43,42)=180
Rollout 91, Act: 2.365
Ġ=Ġ(2,Ġ0)ĊĊSoĠtheĠangleĠbetweenĠBAĠandĠBCĠisĠangleĠatĠB.ĊĊSoĠlet
Rollout 20, Act: 2.348
Ġ-Ġ1.ĠSo,ĠmeaningĠthatĠtheĠleastĠcommonĠmultipleĠofĠallĠpossibleĠordersĠ(i.e.,Ġthe
Top 16 Negative Activations
Rollout 20, Act: -2.367
,Ġp^{kn}Ġâī¡Ġ1ĠmodĠ47ĠforĠallĠprimesĠpĠâīłĠ47
Rollout 66, Act: -2.369
Ġparameterization,ĠsometimesĠyouĠhaveĠmĠandĠnĠcoprimeĠbutĠleadĠtoĠaĠscaledĠtriplet.ĠWait,
Rollout 66, Act: -2.374
Ġn2.ĠSinceĠtheĠlegsĠneedĠtoĠbeĠcoprime,ĠwhichĠoccursĠwhenĠmĠandĠnĠareĠcop
Rollout 66, Act: -2.384
ĠcĠ=Ġp.ĊĊSinceĠtheĠsidesĠareĠpairwiseĠcoprime,Ġit'sĠaĠprimitiveĠPythagoreanĠtriple
Rollout 64, Act: -2.385
)/2Ġ\),ĠthenĠforĠ\(ĠAĠ\)ĠandĠ\(ĠBĠ\)ĠtoĠbeĠintegersĠ(toĠensure
Rollout 71, Act: -2.392
ĠSimilarĠforĠk2.ĠSinceĠ3c2ĠandĠk2ĠsumĠtoĠ8044Ġ(
Rollout 66, Act: -2.429
rimeĠwithĠboth.ĠHenceĠtheĠlegsĠareĠcoprimeĠonlyĠifĠtheyĠareĠprimitive.ĠButĠifĠthe
Rollout 21, Act: -2.431
1.ĠButĠifĠf(x)ĠmustĠmapĠtoĠZ,ĠthenĠifĠxĠisĠincreasedĠbyĠ1,
Rollout 71, Act: -2.486
c2Ġ+8044ĠisĠaĠperfectĠsquare.ĊĊGivenĠthatĠ8044Ġ-3
Rollout 66, Act: -2.490
ĠofĠprimitiveĠtriples,ĠmĠandĠnĠmustĠbeĠcoprime.ĠTherefore,ĠtoĠhaveĠgcd(m,Ġn
Rollout 66, Act: -2.582
rimeĠwithĠeachĠlegĠifĠtheĠlegsĠareĠcoprime.ĠSoĠsinceĠhypotenuseĠcĠ=Ġm
Rollout 83, Act: -2.610
i+3}Ġâī¡Ġ0Ġ(modĠ3)ĊĊSubtractingĠtheĠfirstĠequationĠfromĠthe
Rollout 89, Act: -2.677
ĠTherefore,ĠtheĠtotalĠnumberĠofĠchildrenĠmustĠbeĠaĠmultipleĠofĠ5,ĠasĠ1/5th
Rollout 71, Act: -2.688
44Ġ-3c2ĠtoĠbeĠaĠperfectĠsquare.ĊĊAlternatively,ĠsinceĠ8044Ġitself
Rollout 66, Act: -2.874
.ĠWait,Ġno,ĠallĠthreeĠareĠpairwiseĠcoprime?ĠWait,Ġno.ĠWait,Ġhypoten
Rollout 66, Act: -2.958
ĠprimitiveĠPythagoreanĠtriple,ĠtwoĠlegsĠareĠcoprime,ĠandĠhypotenuseĠisĠnotĠcopr

Layer 28

GATE_PROJ

Top 16 Positive Activations
Rollout 30, Act: 3.578
ĠproblemĠasksĠtoĠ"evaluateĠtheĠinfiniteĠsum",ĠlikelyĠasĠaĠfraction.ĠButĠlet'sĠconfirm.ĊĊTherefore,
Rollout 3, Act: 3.459
olarĠmassĠandĠR?ĠNo,ĠtheĠquestionĠisĠtoĠcomputeĠtheĠspeed,ĠsoĠtheyĠmustĠassumeĠaĠtemperature
Rollout 33, Act: 3.452
ĠthanĠkĠinĠtheĠfirstĠkĠpositions,ĠwhichĠisĠequivalentĠtoĠsayingĠthatĠtheĠfirstkĠpositionsĠcannotĠbeĠa
Rollout 96, Act: 3.420
ĠtranslationĠandĠrotation.ĊĊWait,ĠbecauseĠtheĠcoordinateĠsystemsĠareĠrotatedĠandĠshifted.ĠTheĠoriginalĠcoordinateĠsystemĠisĠtransformed
Rollout 25, Act: 3.411
ĠmakeĠcontactĠwithĠtheĠEarth?"ĊĊSo,ĠtheĠquestionĠis:ĠgivenĠtheĠsameĠapproachĠspeedĠatĠinfinityĠandĠthe
Rollout 22, Act: 3.381
333ĠâīĪĠ?ĊĊWeĠcanĠwriteĠthatĠasĠ18,168.77
Rollout 71, Act: 3.352
Ġ-2011,ĠwhichĠcanĠbeĠwrittenĠasĠabĠ+Ġc(aĠ+Ġb)Ġ=Ġ-2
Rollout 26, Act: 3.329
.ĊĊButĠhowĠisĠtheĠquestionĠphrased?Ġ"theĠcalculatedĠstandardĠmolarĠenthalpy...Ġat
Rollout 99, Act: 3.327
ĠWhat'sĠtheĠdecimalĠvalueĠofĠx3?"ĠWhichĠisĠconfusing,ĠsinceĠtwoĠiterationsĠwouldĠnormallyĠgetĠtoĠx
Rollout 33, Act: 3.310
ĠnumberĠgreaterĠthanĠkĠinĠtheĠfirstĠkĠpositions,ĠwhichĠisĠequivalentĠtoĠsayingĠthatĠtheĠfirstkĠpositionsĠcannot
Rollout 10, Act: 3.308
Ġexist.ĠAsĠaĠcontestĠproblem,ĠtheĠsolutionĠmayĠbeĠfoundĠinĠrealizingĠtheĠperiodicityĠfailed,ĠbutĠthat
Rollout 25, Act: 3.303
ĠcontactĠwithĠtheĠEarth?"ĊĊSo,ĠtheĠquestionĠis:ĠgivenĠtheĠsameĠapproachĠspeedĠatĠinfinityĠandĠtheĠsame
Rollout 88, Act: 3.294
ĠtheĠentireĠactivationĠoutputĠofĠthisĠfirstĠlayer",Ġi.e.,ĠifĠyouĠflattenĠallĠfiltersĠandĠchannelsĠintoĠa
Rollout 14, Act: 3.271
ĠsideĠisĠ$I^3$,ĠperhapsĠwritingĠthatĠasĠ$IĠ\cdotĠIĠ\cdotĠI$,Ġand
Rollout 0, Act: 3.261
Ġ20!.ĊĊHence,ĠtheĠquestionĠisĠequivalentĠto:ĠhowĠmanyĠpairsĠ(a,Ġb)
Rollout 94, Act: 3.254
)ĠisĠnotĠentangled.ĊĊSoĠtheĠquestionĠisĠ"whichĠsetĠofĠstatesĠmentionedĠbelowĠareĠonlyĠentangled
Top 16 Negative Activations
Rollout 70, Act: -1.342
0}^kĠd_iĠ*Ġ10^iĠ=Ġ2Ġ*Ġsum_{i=0
Rollout 70, Act: -1.345
0}^kĠd_iĠ\cdotĠ7^iĠ\).ĊĠĠĠ-ĠTheĠbase-10
Rollout 70, Act: -1.364
=0}^kĠd_iĠ(10^iĠ-Ġ2*7^i)Ġ=
Rollout 70, Act: -1.382
ĠaĠbase-10Ġnumber,ĠequalĠtwiceĠ\(ĠNĠ\).ĠToĠfindĠtheĠlargestĠsuchĠ\(ĠNĠ\
Rollout 96, Act: -1.382
87/28561)Ġ+5s=Ġ-4106/16
Rollout 70, Act: -1.401
=2N.ĊĊThus,ĊĊd3*1000Ġ+Ġd2*100Ġ+
Rollout 70, Act: -1.406
0}^kĠd_iĠ\cdotĠ10^iĠ\).ĊĠĠĠ-ĠWeĠneedĠ\(ĠD
Rollout 70, Act: -1.439
2N.ĊĊThus,ĊĊd3*1000Ġ+Ġd2*100Ġ+d
Rollout 70, Act: -1.473
=0}^kĠd_iĠ(10^iĠ-Ġ2Ġ\cdotĠ7^i
Rollout 70, Act: -1.533
0}^kĠd_iĠ*Ġ10^i.ĠSo,ĠweĠneedĠtheĠsameĠdigitsĠd
Rollout 70, Act: -1.549
_0.ĠThenĠD=d_k*10^kĠ+...+d_1*10
Rollout 70, Act: -1.596
d0.ĊĊD=Ġd3*10^3Ġ+d2*10^2Ġ+
Rollout 70, Act: -1.626
2N.ĊĊTherefore,ĊĊd_k*10^kĠ+Ġ...Ġ+Ġd_1*10
Rollout 70, Act: -1.627
}^kĠd_iĠ\cdotĠ10^iĠ=Ġ2Ġ\sum_{i=0
Rollout 70, Act: -1.634
_0Ġ=Ġ2*(d_k*7^kĠ+Ġ...Ġ+Ġd_1*7Ġ+
Rollout 70, Act: -1.854
}^kĠd_iĠ\cdotĠ10^iĠ\).ĊĠĠĠ-ĠWeĠneedĠ\(ĠDĠ=

UP_PROJ

Top 16 Positive Activations
Rollout 36, Act: 3.159
ĠLet'sĠcomputeĠ18,8642:ĊĊFirstĠcomputeĠ18,000
Rollout 4, Act: 3.084
000,ĠsinceĠ5025*1000=5,025
Rollout 36, Act: 3.076
58.12ĠbecauseĠ1582=24,964ĠandĠ1
Rollout 36, Act: 3.036
2):ĊĊSinceĠ19,0002Ġ=Ġ361,000,
Rollout 36, Act: 2.897
0ĊĊComputeĠ19,3602=?ĊĊ19,360^2
Rollout 71, Act: 2.878
ĠdĠ<=Ġ25Ġ(sinceĠ252=625ĠandĠ262=
Rollout 36, Act: 2.824
2=?ĊĊ19,360^2=Ġ(19,000Ġ+Ġ
Rollout 36, Act: 2.789
ĠtoĠ75.0ĠbecauseĠ752=5,625,ĠveryĠclose.ĊĊ
Rollout 22, Act: 2.773
ĠcomputeĠthis.ĊĊSinceĠ1,7002Ġ=Ġ2,890,00
Rollout 71, Act: 2.754
ĠbeĠupĠtoĠ25ĠsinceĠ252=625Ġ<=670.3
Rollout 11, Act: 2.731
Ġe.g.,ĠkĠ=Ġ2iĠ(modulusĠ2),ĠthenĠcĠ=1/(Ġ(2
Rollout 38, Act: 2.725
27ĠbecauseĠ527*527=?ĊĊWell,Ġ5002=
Rollout 71, Act: 2.716
Ġ=>Ġc>=45Ġ(sinceĠ442=1936,Ġ452
Rollout 38, Act: 2.700
055.ĠComputeĠ10542=(1055-1)^2=
Rollout 71, Act: 2.692
4.81,Ġno.Ġ442=1936,Ġ452
Rollout 22, Act: 2.683
0).ĠĊĊSinceĠ1,6942Ġ=Ġ2,869,63
Top 16 Negative Activations
Rollout 32, Act: -1.487
pmatrix}Ġ0Ġ&Ġ1Ġ\\Ġ-\omega^2Ġ&Ġ-\omegaĠ\end{
Rollout 28, Act: -1.488
ĠĠĠBĠhĠ+Ġ2CĠkĠ-ĠCĠbĠ=Ġ0ĊĠĠĠ\end{cases}Ċ
Rollout 51, Act: -1.492
ĠintegralĠbecomes,ĠafterĠvariableĠshiftingĠyĠ=xĠ-θ,ĠintegralĠbecomes:ĊĊâĪ«02ÏĢĠsin8
Rollout 43, Act: -1.499
Ġ|0\rangle\)ĊĠĠĠ-Ġ\(-\gamma|1\rangleĠ\otimesĠ|0
Rollout 28, Act: -1.515
=0ĊĊFactorĠtermsĠinĠ(2kĠ-b)ĠandĠ(2hĠ-a):ĊĊ(2
Rollout 28, Act: -1.533
Ġ4a2y2Ġ-4a2bĠyĠ=4a2y(yĠ-bĠ)ĊĊ
Rollout 28, Act: -1.535
ĠbĠ=Ġ0ĠâĩĴĠC(2kĠ-Ġb)Ġ+ĠBĠhĠ=Ġ0ĊĊSoĠwe
Rollout 28, Act: -1.540
0Ġ...Ġ(1)ĊĊC(2kĠ-Ġb)Ġ+ĠBĠhĠ=Ġ0Ġ...Ġ(2
Rollout 64, Act: -1.544
ĠtermsĠareĠstillĠintegers.ĠTherefore,ĠtheĠonlyĠconstraintsĠareĠthatĠa1ĠandĠa2ĠareĠpositiveĠintegers.ĊĊ
Rollout 12, Act: -1.559
Ġ=Ġ[ĠtĠ/Ġsqrt(Ġ(t2Ġ-Ġν2)(1Ġ-Ġt^2)Ġ)Ġ]
Rollout 75, Act: -1.581
ĠshouldĠbeĠpossible.ĠWait,ĠtheĠonlyĠconstraintĠhereĠisĠthatĠSĠâī¡Ġ3Ġmod4.ĠLet
Rollout 28, Act: -1.592
3(uĠ-2)^2Ġ-12Ġ+Ġv2Ġ+9=0ĠâĨĴĠ3(uĠ-
Rollout 12, Act: -1.607
Ġ[ĠtĠ/Ġsqrt(Ġ(t^2Ġ-Ġν^2)(1Ġ-Ġt^2)Ġ)
Rollout 28, Act: -1.610
2Ġ+Ġ2a2ĠbĠkĠ-4Ġb2Ġh2Ġ+Ġ2aĠb2Ġh
Rollout 28, Act: -1.617
Ġ(-2AĠa^2Ġ-Ġ2CĠb^2)Ġ+ĠBĠabĠ=Ġ0Ġ\
Rollout 45, Act: -1.767
4Ġ=Ġf(-1)Ġ+Ġ5Ġ-ĠbĠ=>Ġf(-1)Ġ=Ġ-4Ġ-5

DOWN_PROJ

Top 16 Positive Activations
Rollout 60, Act: 2.870
Ġ6ĠHz.ĊĊButĠwhyĠtheĠinitialĠapproximateĠcalculationĠgaveĠsomethingĠlikeĠ5.92ĠHz,Ġbut
Rollout 23, Act: 2.854
5ĠBq,ĠwhichĠisĠoptionĠ(D)Ġ117.555,ĠcloseĠto
Rollout 70, Act: 2.670
153.ĊĊButĠN=35mĠ+13.ĠThus,ĊĊ105d1
Rollout 2, Act: 2.658
109ĠinĠbothĠcases.ĠSoĠTheĠanswerĠis109.ĠHuh.ĊĊButĠwait,
Rollout 36, Act: 2.623
0ĊĊCompareĠtoĠourĠÎĶ=375,662,592.ĠIt's
Rollout 36, Act: 2.582
00ĊĊCompareĠtoĠourĠÎĶ=375,662,592.ĠIt
Rollout 39, Act: 2.567
4.ĠHmmm.Ġ27Ġisn'tĠ4,ĠsoĠseemsĠoff.ĠSoĠIĠmustĠbe
Rollout 91, Act: 2.539
ĠC|=2ĠregardlessĠofĠt?ĊĊSoĠtheĠanswerĠisĠ2?ĊĊButĠwait,Ġwait.ĠSoĠaccording
Rollout 87, Act: 2.513
62.ĠThat'sĠnotĠmatchingĠ3.795.ĊĊAlternatively,ĠdifferentĠcombination.ĊĊTestĠanother
Rollout 30, Act: 2.474
5Ġ*2014Ġ)ĊĊSoĠtheĠanswerĠisĠ1Ġ+Ġ1/(2015
Rollout 58, Act: 2.449
7171.ĠThus,ĠtheĠanswerĠshouldĠbeĠ\boxed{2037171
Rollout 11, Act: 2.417
]Ġ+Ġ[Ġr2Ġ+Ġ2/rĠcosθĠ]=Ġ2rĠcosθĠ+1/r
Rollout 70, Act: 2.410
102.ĊĊButĠN=35mĠ+32.ĊĊThus,ĊĊ105d1
Rollout 8, Act: 2.410
.246.ĊĊThus,ĠtheĠanswerĠwouldĠbeĠ31Ġ+126=15
Rollout 90, Act: 2.402
ĠtheĠdeducedĠanswer,Ġyes.ĠSoĠtheĠanswerĠisĠ\boxed{1008919
Rollout 38, Act: 2.384
24=61.ĊĊTherefore,ĠtheĠanswerĠisĠ\boxed{61}ãĢĤĊĊ**FinalĠAnswer
Top 16 Negative Activations
Rollout 89, Act: -2.458
Wait,ĠwaitĠaĠminute.ĠLetĠmeĠreadĠtheĠproblemĠagain.Ġ"ExactlyĠ2/5thĠof
Rollout 81, Act: -2.461
/121Ġ-2601/121=Ġ(6171Ġ-2
Rollout 96, Act: -2.464
*37=481ĠâĨĴTotalĠ109,200Ġ+481
Rollout 9, Act: -2.468
Ġ(10Ġ-5)/(10Ġ-Ġ(10Ġ-Ġh))Ġ=Ġ5/h.ĊĊTherefore
Rollout 81, Act: -2.471
,ĠpossiblyĠerrorĠinĠmethodsĠcomplicated.ĊĊLetĠmeĠstepĠback.ĠIfĠbothĠapproachesĠcorrectlyĠmodelĠtan(theta)=2
Rollout 99, Act: -2.474
Ġ(-4)/(-3.456315)ĠâīĪĠ1.157
Rollout 99, Act: -2.496
ĠinĠtheĠsignĠforĠtheĠdenominator.ĠLetĠmeĠgoĠbackĠstepĠbyĠstep.ĊĊSo,ĠstepĠ1:
Rollout 79, Act: -2.503
ĠNo.ĊĊThisĠisĠreallyĠfrustrating.ĠWait,ĠgoingĠbackĠtoĠanagramĠofĠ"minorĠchanges":ĠM,
Rollout 59, Act: -2.517
/28,ĊĊP(X=3)Ġ=Ġ1/7Ġ=Ġ4/28.ĊĊ
Rollout 87, Act: -2.539
ĠisĠintegratingĠ1/sqrt(ξ2Ġ+Ġη2Ġ+ĠÎ2})ĠoverĠtheĠunit
Rollout 56, Act: -2.575
44ĊĊThusĠsqrt[196/144Ġ+Ġ363/14
Rollout 1, Act: -2.594
2)ĠdĠâīĪĠ1.414d.ĠTheĠmutualĠdistanceĠbetweenĠcentersĠisĠd.
Rollout 22, Act: -2.632
ĠnumberĠ(6.02214076e23Ġmolecules/mol):ĊĊ0.0
Rollout 70, Act: -2.673
47=6.ĠThenĠ6ĠdividedĠbyĠ7=0,ĠsoĠdigitsĠshouldĠbeĠ3
Rollout 61, Act: -2.673
0}Ġ\)ĠwhoseĠbinaryĠexpansionsĠhaveĠexactlyĠtwoĠ1's,ĠweĠneedĠtoĠfindĠtheĠprobabilityĠthat
Rollout 97, Act: -2.715
Ġ2ĠmolesĠofĠN2O4ĠandĠ1ĠmoleĠofĠO2ĠareĠformed.ĊĊSo,

Layer 29

GATE_PROJ

Top 16 Positive Activations
Rollout 22, Act: 4.097
=ĠthatâĢĻsĠnegative,Ġmaybe.ĊĊWait,ĠbetterĠway:Ġ145,458.
Rollout 20, Act: 4.040
ĠBecause4^7=,waitĠno,ĠbetterĠway.ĠWait,2^14âī¡4Ġmod
Rollout 73, Act: 3.997
n+1}ĊĊWait,ĠperhapsĠaĠdifferentĠway:ĊĊLookingĠtermĠbyĠterm:ĊĊFirstĠtermĠofĠ2
Rollout 41, Act: 3.995
ĠfirstĠdigit).ĠWait,Ġno.ĊĊWait,ĠperhapsĠbetterĠtoĠsplitĠintoĠtwoĠcases:ĠwhenĠoneĠofĠthe
Rollout 20, Act: 3.957
.ĠBecause4^7=,waitĠno,ĠbetterĠway.ĠWait,2^14âī¡4
Rollout 13, Act: 3.917
Ġapproach.ĠWait,ĠmaybeĠIĠshouldĠtryĠdirectĠmultiplication:ĊĊ1.2544Ċ*1.
Rollout 70, Act: 3.888
^{-1}*2Ġmod5.ĠWait,ĠperhapsĠbetterĠtoĠuseĠpreviousĠequations.ĊĊSo,ĠifĠd0
Rollout 72, Act: 3.862
Ġ2.ĠNo,ĠbetterĠtoĠuseĠexactĠformula.Ġcos(72°)Ġ=Ġ(sqrt(
Rollout 22, Act: 3.847
ĠthatâĢĻsĠnegative,Ġmaybe.ĊĊWait,ĠbetterĠway:Ġ145,458.9
Rollout 25, Act: 3.845
ĠmeĠsquareĠbothĠsidesĠfirst.ĠWait,ĠperhapsĠaĠbetterĠsubstitution.ĊĊLetâĢĻsĠletĠxĠ=Ġr2,
Rollout 96, Act: 3.838
2xĠ+5y=90ĊĊSolvingĠforĠvariables:ĊĊAlternatively,ĠsolveĠforĠoneĠvariableĠfrom
Rollout 8, Act: 3.834
BB...ĠWait,ĠGGRRĠwouldĠactuallyĠbeĠsimilar:ĠP(5,2)*P(5,
Rollout 40, Act: 3.829
ī^n,ĊĊWait,ĠletĠmeĠgroupĠtheĠterms:ĊĊFirstĠterm:Ġ5ĠÏī^nĊĊSecond
Rollout 9, Act: 3.795
ĠdxĊĊWait,ĠwaitĠsubtractĠtheĠlowerĠfromĠupper:ĊĊ=ĠâĪ«[0Ġto10]Ġ[
Rollout 36, Act: 3.769
Wait,Ġwait,ĠholdĠon:ĊĊWait,ĠmoreĠaccurately,Ġ(s2Ġ+Ġ1886
Rollout 16, Act: 3.760
ĠsaplingĠisĠnotĠyetĠconverted.ĊĊWait,ĠperhapsĠanotherĠapproach.ĠSinceĠeachĠday,ĠeachĠremainingĠsapling
Top 16 Negative Activations
Rollout 77, Act: -1.709
,52)$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠproblemĠhereĠwhereĠI
Rollout 12, Act: -1.712
P.ĠTuran]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠproveĠthatĠtheĠfunction
Rollout 47, Act: -1.713
ĠofĠ$N$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠthere'sĠthisĠproblemĠaboutĠaĠtrain
Rollout 89, Act: -1.722
ĠareĠinĠtheĠclass?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠseeĠhere.ĠSoĠtheĠproblem
Rollout 1, Act: -1.723
cal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠthereĠexists
Rollout 74, Act: -1.731
ĠandĠ$\theta$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠderiveĠanĠexpressionĠfor
Rollout 94, Act: -1.743
(D)Ġa,bĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠdetermineĠwhichĠofĠthese
Rollout 97, Act: -1.745
mathrm{Y}$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠhaveĠthisĠdecomposition
Rollout 80, Act: -1.764
ĠofĠ$N$Ġ?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthisĠproblem
Rollout 29, Act: -1.764
rhoĠgH$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠthisĠphysics
Rollout 42, Act: -1.769
-5-eneĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠtheĠproduct
Rollout 44, Act: -1.774
{T}_{2}$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠwhichĠof
Rollout 27, Act: -1.783
ĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthisĠproblemĠhereĠwhere
Rollout 23, Act: -1.798
55ĠBqĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblemĠstepĠbyĠstep
Rollout 57, Act: -1.806
.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠifĠwe
Rollout 98, Act: -1.806
(D)Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigureĠoutĠhowĠmany

UP_PROJ

Top 16 Positive Activations
Rollout 37, Act: 2.922
5ĠandĠtheirĠmultiplicities.ĠButĠperhapsĠforĠeachĠprimeĠpĠĠâī¤Ġ2015,Ġthe
Rollout 37, Act: 2.799
ĠseemsĠcomplicatedĠifĠweĠneedĠtoĠcheckĠitĠforĠeachĠprime.ĊĊAlternatively,ĠmaybeĠtheĠdeterminantĠcanĠbeĠexpressedĠas
Rollout 33, Act: 2.794
ĠfirstĠkĠtermsĠincludeĠatĠleastĠoneĠnumberĠgreaterĠthanĠk.ĠLetĠmeĠtryĠtoĠunpackĠthisĠproblemĠandĠfigure
Rollout 93, Act: 2.775
,ĠbutĠtheĠpowerĠsumĠsymmetricĠpolynomialsĠofĠtheĠroots.ĠĠIfĠx_kĠ=Ġcos(ÏĢk
Rollout 90, Act: 2.768
'sĠaĠcorrespondingĠNĠâī¡Ġ-897MĠmodĠ1009.ĊĊButĠNĠis
Rollout 37, Act: 2.691
Ġâī¤Ġ2015,ĠtheĠexponentĠofĠpĠinĠdet(A)ĠisĠatĠleastĠtheĠexponentĠof
Rollout 37, Act: 2.686
ĠforĠanyĠprimeĠpĠâī¤Ġ2015,ĠtheĠexponentĠofĠpĠinĠdet(A)ĠisĠat
Rollout 66, Act: 2.677
Ġm2Ġ-Ġn2,ĠbĠ=Ġ2mn,ĠcĠ=Ġm2Ġ+Ġn2.ĠLet
Rollout 0, Act: 2.651
Ġb\).ĠSoĠforĠeachĠsuchĠcoprimeĠpairĠwhereĠ\(aĠ<Ġb\),ĠweĠhaveĠa
Rollout 33, Act: 2.644
ĠshouldĠbeĠatĠleastĠoneĠnumberĠthat'sĠbiggerĠthanĠk.ĠAndĠthisĠconditionĠneedsĠtoĠholdĠforĠallĠk
Rollout 20, Act: 2.640
Ġ43.ĊĊSo,ĠweĠneedĠforĠeachĠprimeĠp,ĠandĠeachĠkĠâī¥Ġ1,ĠÏĥ
Rollout 20, Act: 2.610
isorsĠofĠa.ĠSo,ĠifĠaĠisĠaĠprimeĠp,ĠÏĥ(p)Ġ=Ġ1Ġ+Ġp
Rollout 44, Act: 2.604
Ġit'sĠisothermal,ĠtheĠworkĠwouldĠbeĠnRTĠln(V1Ġ/ĠV2).ĠOr,Ġif
Rollout 37, Act: 2.600
orem,ĠitĠsufficesĠtoĠshowĠthatĠforĠanyĠprimeĠpĠâī¤Ġ2015,ĠtheĠexponent
Rollout 33, Act: 2.588
ĠnumbersĠ1ĠthroughĠ6ĠsuchĠthatĠforĠeachĠkĠfromĠ1ĠtoĠ5,ĠtheĠfirstĠk
Rollout 93, Act: 2.585
Wait,ĠanotherĠthought:ĠwhenĠdealingĠwithĠsumsĠoverĠrootsĠofĠunityĠorĠtrigonometricĠsums,ĠsometimesĠsymmetry
Top 16 Negative Activations
Rollout 53, Act: -2.847
ĠbeĠanswer.ĊĊHence,ĠtheĠareaĠisĠ\boxed{\dfrac{27}{2}}.ĊĊ
Rollout 66, Act: -2.848
=182.ĠThus,ĠanswerĠisĠ\boxed{182}.ĊĊTherefore,ĠdespiteĠm
Rollout 13, Act: -2.860
7.12Ċ\]ĊĊ\[Ċ\boxed{187.12}Ċ\]
Rollout 8, Act: -2.861
Ġ=Ġ157\).ĊĊ\[Ċ\boxed{157}Ċ\]<|im_end|>
Rollout 43, Act: -2.862
ĠbeĠ\boxed{C}.ĊĊĊ**FinalĠAnswer**Ċ\boxed{C}ĊĊ<|im_start|>answerĊTheĠproblem
Rollout 50, Act: -2.863
112\).ĊĊ**FinalĠAnswer**Ċ\boxed{112}ĊĊ<|im_start|>answerĊGivenĠthe
Rollout 5, Act: -2.872
5ĠisĠ181.ĊĊ\[Ċ\boxed{181}Ċ\]<|im_end|>
Rollout 12, Act: -2.895
Ġ\nuĠ<Ġ1Ġ\).ĊĊ\[Ċ\boxed{f(\nu)Ġ\text{ĠisĠmon
Rollout 68, Act: -2.898
Ġ=Ġ501Ġ\).ĊĊ\[Ċ\boxed{501}Ċ\]<|im_end|>
Rollout 32, Act: -2.921
ĠisĠdefinitelyĠ4018.ĊĊThereforeĠ\boxed{4018}.ĊĊ**FinalĠAnswer
Rollout 76, Act: -2.927
Ġ=Ġ929\).ĊĊ\[Ċ\boxed{929}Ċ\]<|im_end|>
Rollout 78, Act: -2.956
ĠnĠ=Ġ49Ġ\).ĊĊ\[Ċ\boxed{49}Ċ\]<|im_end|>
Rollout 16, Act: -2.991
9953Ċ\]ĊĊ\[Ċ\boxed{789953}Ċ\]
Rollout 50, Act: -3.045
Ġ112Ċ\]ĊĊ\[Ċ\boxed{112}Ċ\]<|im_end|>
Rollout 61, Act: -3.057
80Ġ=Ġ913Ġ\).ĊĊ\boxed{913}<|im_end|>
Rollout 59, Act: -3.113
Ġ=Ġ515\).ĊĊ\[Ċ\boxed{515}Ċ\]<|im_end|>

DOWN_PROJ

Top 16 Positive Activations
Rollout 14, Act: 2.900
ĠtheĠ1/xĠintegralĠbetweenĠ\ĠtheĠtails.ĊĊAlternatively,ĠuseĠHölderĠinequalityĠinĠtheĠform:ĊĊ(
Rollout 87, Act: 2.815
ĠbyĠsomething.ĠNotĠsure,ĠbutĠsomeĠcombination.ĊĊAlternatively,ĠrecallĠthatĠpotentialĠatĠcenterĠisĠdifferentâĢĶtheĠintegral
Rollout 79, Act: 2.803
ĠanagramĠofĠ"monarchesign"?ĠNo.ĠAlternatively,Ġ"morchaseingn"ĠâĢĵĠwhich
Rollout 46, Act: 2.796
,ĠwhileĠweĠareĠaskedĠforĠsecondĠtimeĠderivatives.ĊĊAlternatively,ĠapplyingĠUĠinĠcontextĠofĠLagrangianĠformal
Rollout 49, Act: 2.792
IfĠAĠisĠwell-orderedĠbyĠsomeĠrelation,ĠbutĠneedĠtoĠgetĠaĠsequence,ĠbutĠIĠcan't
Rollout 37, Act: 2.764
ĠifĠa1ĠandĠa2ĠareĠconsecutiveĠintegers,ĠbutĠweĠhaveĠnoĠinfo.ĊĊWait,Ġlet'sĠplug
Rollout 79, Act: 2.747
'tĠpossible.ĠĊĊNext,Ġ"Chancers"ĠneedsĠtwoĠC's.ĠNo.ĠĊĊNext,Ġ"
Rollout 49, Act: 2.738
ityĠwithĠhereditaryĠLindelöfness?ĊĊAlternatively,ĠconsiderĠasĠXĠisĠhereditarilyĠLindel
Rollout 87, Act: 2.737
ĠmultipleĠintegrationsĠsubstitutingĠhyperbolicĠsubstitutions.ĊĊAlternatively,ĠinĠFeynman'sĠlectures,ĠtheyĠsolve
Rollout 37, Act: 2.734
ĠofĠfiniteĠdifferencesĠbutĠIĠdon'tĠrememberĠwell.ĊĊAlternatively,ĠseeingĠthatĠforĠthe.ĠstatementĠtoĠbeĠtrue
Rollout 87, Act: 2.723
.ĠTheĠcubeĠhasĠedges,Ġcorners,Ġetc.ĊĊAlternatively,ĠintegrateĠinĠsphericalĠcoordinatesĠbutĠadjustingĠforĠtheĠcube
Rollout 46, Act: 2.713
ī2ĠxĠ/Ġsomething...ĠWait,Ġno.ĊĊAlternatively,ĠperhapsĠconfusionĠbetweenĠLaplace'sĠandĠgradient.
Rollout 93, Act: 2.710
ĠdirectlyĠbecauseĠtheĠrelationsĠdependĠonĠparityĠofĠk.ĊĊAlternatively,ĠforĠallĠx_kĠweĠhaveĠT_{1
Rollout 87, Act: 2.698
ĠresultĠinĠtermsĠwithĠinverseĠtrigonometricĠfunctions.ĊĊAlternatively,ĠsinceĠeachĠstepĠinvolvesĠmultipleĠintegrationsĠsubstituting
Rollout 79, Act: 2.687
,Ġ"Mechanic'sĠSign"ĠâĢĵĠno.ĊĊAlternatively,ĠmaybeĠ"MinorĠchanges"ĠbecomesĠ"ChangeĠminor
Rollout 79, Act: 2.686
".ĊĊAlternatively,ĠthinkĠofĠ"Mosaic"ĠâĢĵĠbutĠnotĠenoughĠletters.ĊĊAlternatively,ĠmaybeĠtheĠ4
Top 16 Negative Activations
Rollout 41, Act: -2.156
ĠNumbersĠwhereĠbothĠdigitsĠareĠnon-zero.ĠThen,ĠweĠcanĠarrangeĠthoseĠtwoĠdigitsĠfreely,ĠbutĠensuringĠno
Rollout 41, Act: -2.171
ĠtheĠtotalĠnumberĠaccordingĠtoĠtheĠalternativeĠapproachĠ(ifĠleadingĠzerosĠareĠallowed)ĠisĠ640-
Rollout 41, Act: -2.179
Ġ3Ġnumbers.ĠTherefore,ĠtotalĠnumbersĠforĠthisĠcaseĠwouldĠbeĠ9Ġ*Ġ3Ġ=Ġ2
Rollout 77, Act: -2.190
6)=12,ĠwhichĠisĠLCM(4,6)=12.ĠIndeed.ĠThenĠf
Rollout 32, Act: -2.190
ī2Ġ-ĠÏī)=4Ġ(sameĠasĠaboveĠsinceĠ1Ġ-ĠÏī2Ġ-ĠÏī
Rollout 41, Act: -2.195
ĠtwoĠdifferentĠdigitsĠisĠ243,ĠplusĠtheĠnumbersĠwithĠallĠthreeĠdigitsĠtheĠsame,ĠwhichĠis
Rollout 94, Act: -2.215
able.ĊĊHence,Ġagain,ĠmyĠpreviousĠconclusionĠholds.ĊĊTherefore,ĠtheĠcorrectĠanswerĠisĠoptionĠ(B
Rollout 39, Act: -2.215
10.ĠButĠinĠthisĠsmallerĠcase,ĠhowĠwouldĠitĠbreakĠdownĠwithĠsubtraction?ĠWell,ĠtakingĠus
Rollout 3, Act: -2.219
ĠWait,ĠbutĠIĠmightĠhaveĠtoĠverifyĠwhichĠoneĠisĠtheĠactualĠmeanĠspeed.ĠLetĠmeĠthink.ĊĊMean
Rollout 60, Act: -2.239
5.92ĠHz,ĠbutĠusingĠexactĠfractionsĠgivesĠexactlyĠ6ĠHz?ĠProbablyĠbecauseĠIĠapproximated
Rollout 48, Act: -2.245
16=7ĠwithĠcarryĠ1.ĠSoĠweĠwriteĠ7.ĊĊNextĠ(tensĠdigitĠof
Rollout 95, Act: -2.247
.ĠThereĠisĠaĠdiscrepancyĠhere.ĠSoĠwhichĠisĠcorrect?ĊĊAlternatively,Ġthere'sĠaĠconfoundment.
Rollout 63, Act: -2.293
1}{2}f(2,2)Ġ\),ĠsubstituteĠintoĠabove:ĊĊ\[Ċf(2
Rollout 32, Act: -2.363
ĠgivesĠ669*6.ĠThenĠtheĠremainingĠtwoĠtermsĠ(MĠandĠM2)ĠwouldĠgive
Rollout 41, Act: -2.401
,ĠplusĠtheĠnumbersĠwithĠallĠthreeĠdigitsĠtheĠsame,ĠwhichĠisĠ9.ĠSoĠinĠtotal,Ġ
Rollout 97, Act: -2.721
5Ġe^{-kt}ĊĊWhichĠisĠtheĠsameĠresultĠasĠwhenĠIĠdidĠitĠviaĠODEĠandĠstoich

Layer 30

GATE_PROJ

Top 16 Positive Activations
Rollout 41, Act: 2.478
,000ĠthatĠhaveĠatĠmostĠtwoĠdifferentĠdigitsĠisĠ\boxed{927}.ĊĊ<|im_start|>
Rollout 41, Act: 2.246
ĠĠĠ-Ġ**CaseĠ2**:ĠExactlyĠtwoĠdifferentĠdigits.ĊĠĠĠĠĠ-Ġ**SubcaseĠA**:ĠNeitherĠdigit
Rollout 41, Act: 2.246
0,000ĠhaveĠatĠmostĠtwoĠdifferentĠdigits,ĠweĠconsiderĠnumbersĠofĠdifferentĠlengthsĠ(1-digit
Rollout 41, Act: 2.220
0,000ĠhaveĠatĠmostĠtwoĠdifferentĠdigits?ĊĊSoĠifĠIĠjustĠthinkĠofĠallĠnumbersĠfrom
Rollout 41, Act: 2.168
ĠtoĠcountĠthoseĠnumbersĠthatĠuseĠatĠmostĠtwoĠdifferentĠdigits.ĠForĠexample,Ġ111ĠhasĠone
Rollout 41, Act: 2.148
10Ġnumbers.ĊĊForĠnumbersĠwithĠexactlyĠtwoĠdifferentĠdigits:ĠChooseĠtwoĠdigitsĠ(C(10,
Rollout 41, Act: 2.084
ĠsameĠ(9Ġnumbers)ĠplusĠnumbersĠwithĠexactlyĠtwoĠdifferentĠdigitsĠ(243)ĠasĠcomputedĠabove.
Rollout 41, Act: 2.077
ĠeachĠofĠtheseĠnumbersĠmustĠhaveĠatĠmostĠtwoĠdifferentĠdigits.ĠSo,ĠnumbersĠlikeĠ111Ġwould
Rollout 41, Act: 2.069
0,000ĠhaveĠatĠmostĠtwoĠdifferentĠdigits?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġso
Rollout 41, Act: 2.068
ĠĠĠ-Ġ**CaseĠ2**:ĠExactlyĠtwoĠdifferentĠdigits.ĊĠĠĠĠĠ-Ġ**SubcaseĠA**:ĠNeitherĠdigit
Rollout 41, Act: 1.977
ĠHowĠmanyĠfour-digitĠnumbersĠhaveĠatĠmostĠtwoĠdifferentĠdigits?ĊĊAgain,ĠweĠsplitĠintoĠcases:ĊĊCaseĠ
Rollout 41, Act: 1.968
Ġ10Ġnumbers.ĊĊForĠnumbersĠwithĠexactlyĠtwoĠdifferentĠdigits:ĠChooseĠtwoĠdigitsĠ(C(10
Rollout 41, Act: 1.923
ĠnumbersĠ(includingĠleadingĠzeros)ĠwhereĠonlyĠtwoĠdistinctĠdigitsĠareĠpresent,ĠthenĠsubtractĠ1Ġ(theĠnumber
Rollout 41, Act: 1.917
0,000ĠhaveĠatĠmostĠtwoĠdifferentĠdigits?ĠHmm,Ġokay.ĠLetĠmeĠtryĠtoĠbreak
Rollout 41, Act: 1.902
ĠleadingĠzeros):ĊĊTheĠtotalĠnumbersĠwithĠatĠmostĠtwoĠdifferentĠdigitsĠwouldĠbe:ĊĊNumberĠofĠmonochromaticĠnumbers
Rollout 41, Act: 1.898
ĠneedĠtoĠcountĠthoseĠnumbersĠthatĠuseĠatĠmostĠtwoĠdifferentĠdigits.ĠForĠexample,Ġ111Ġhas
Top 16 Negative Activations
Rollout 44, Act: -3.950
),ĠthenĠÎĶVĠ=ĠV1Ġ-ĠV2ĠisĠnegative.ĠSoĠworkĠdoneĠonĠtheĠgasĠisĠ-
Rollout 22, Act: -3.956
,ĠtheĠmeanĠmolecularĠspeed,Ġright.ĠTheĠformulaĠIĠrememberĠisĠsomethingĠlikeĠtheĠsquareĠrootĠofĠ(3
Rollout 69, Act: -3.974
^i)]ĊĊIĠneedĠtoĠevaluateĠthatĠinfiniteĠsumĠandĠexpressĠitĠasĠ1/(aĠ+ĠâĪļ
Rollout 81, Act: -3.989
/17Ġ(negativeĠslope).ĠThen,ĠdifferenceĠisĠ(-h/17Ġ-Ġh/3).ĊĊ
Rollout 99, Act: -3.990
3.ĠHmm,Ġokay.ĠĊĊFirst,ĠIĠrememberĠthatĠtheĠMüllerĠmethodĠisĠaĠnumericalĠmethodĠusedĠto
Rollout 74, Act: -4.005
'sĠmagneticĠfieldĠdifferently,ĠcausingĠtheĠcompassĠdeviation.ĊĊToĠmodelĠthis,ĠIĠshouldĠconsiderĠtheĠEarth'sĠmagnetic
Rollout 27, Act: -4.012
ĠstartĠbreakingĠthisĠdown.ĊĊFirst,ĠIĠneedĠtoĠrecallĠwhatĠrmsĠspeedĠmeans.ĠRootĠmeanĠsquareĠspeed,
Rollout 12, Act: -4.020
ĠoutĠhowĠtoĠapproachĠthis.ĠĊĊFirst,ĠIĠrememberĠthatĠaĠfunctionĠisĠmonotonicallyĠdecreasingĠifĠits
Rollout 28, Act: -4.033
ĠbreakĠthisĠdownĠstepĠbyĠstep.ĊĊFirst,ĠIĠrememberĠthatĠaĠconicĠisĠdefinedĠbyĠfiveĠpoints,
Rollout 48, Act: -4.037
ĠAEE0.ĠAlright,Ġfirst,ĠIĠshouldĠrecallĠtheĠstructureĠofĠanĠIPv4ĠheaderĠtoĠparseĠeach
Rollout 54, Act: -4.059
ethylcyclohexanol'sĠpKaâĢĶprobablyĠsimilarĠtoĠcyclohexanol,ĠmaybeĠaĠbit
Rollout 36, Act: -4.067
456Ġ-Ġ55584):ĊĊ147456Ġ-Ġ55
Rollout 17, Act: -4.087
**I**|.ĠĊĊFirst,ĠIĠneedĠtoĠrecallĠtheĠformulaĠforĠtheĠmagneticĠfieldĠofĠaĠdipole
Rollout 3, Act: -4.126
ĠofĠradonĠgas.ĠHmm,Ġfirst,ĠIĠrememberĠthere'sĠaĠformulaĠforĠtheĠrootĠmeanĠsquareĠspeed
Rollout 92, Act: -4.176
01626.ĠSoĠaddingĠtheseĠtwo:Ġ5.405405Ġ-
Rollout 29, Act: -4.243
ĠheightsĠhĠandĠH.ĊĊFirst,ĠIĠneedĠtoĠrecallĠwhatĠgaugeĠpressureĠis.ĠGaugeĠpressureĠisĠtheĠpressure

UP_PROJ

Top 16 Positive Activations
Rollout 55, Act: 3.621
,Ġ(2,1)Ġentry:Ġ0*ĠstuffĠwouldĠbeĠ0.ĠSimilarly,Ġ(2
Rollout 53, Act: 3.598
Ġ/2,0Ġ)ĊĊFirstĠterm:Ġx1Ġy2Ġ=Ġ(-sqrt73Ġ/2)(0
Rollout 86, Act: 3.527
0Ġ)=0.ĊĊAt0:Ġ0*(Ġ-1Ġ+Ġ(1/3)*1Ġ)
Rollout 73, Act: 3.507
4?ĠBecauseĠwhenĠi=0,Ġ0*2^0=0.ĠSoĠsum_{i
Rollout 40, Act: 3.460
1Ġ)].ĊĊThenĠpluggedĠintoĠtermĠ[10/(1Ġ-ĠÏīĠ)]*(1Ġ+i).
Rollout 86, Act: 3.447
Ġ/3)Ġ=0.ĊĊAt0:0*(-1/3)=0.ĊĊThus,Ġuv=
Rollout 33, Act: 3.431
,ĠbutĠifĠyouĠalreadyĠhaveĠtheĠfirstĠtwoĠelementsĠasĠ1ĠandĠ2Ġ(inĠsomeĠorder),
Rollout 53, Act: 3.401
2)(0Ġ)Ġ=Ġ0ĊĊx2Ġy3Ġ=Ġ(Ġ0Ġ)*(18/s
Rollout 52, Act: 3.396
ĠMaybeĠcheckĠtheĠmath.ĊĊExpressĠtheĠsumĠasĠaverageĠ*ĠnumberĠofĠterms.ĠTheĠaverageĠofĠtheĠsequenceĠfrom
Rollout 56, Act: 3.388
implifyĠtermĠbyĠterm:ĊĊFirstĠterm:Ġ0*(Ġ-âĪļ3/2Ġ-ĠâĪļ3/
Rollout 73, Act: 3.383
ĠbyĠterm.ĊĊi=0:Ġ10*1=10ĊĊi=1:9
Rollout 18, Act: 3.362
ĠtakeĠ(k,Ġn)Ġ=Ġa*(0,Ġx)Ġ+Ġk*(1,Ġy)Ġ=
Rollout 86, Act: 3.355
«xĠcosxĠdx.ĊĊEvaluateĠfirstĠterm:Ġ-Ġ(Ġ(pi/2)^2Ġ*Ġcos(pi
Rollout 56, Act: 3.354
168Ġ)ĊĊArea:ĊĊ=Ġ|Ġx3(y4Ġ-Ġy5)Ġ+Ġx4(y5
Rollout 53, Act: 3.353
Ġ*0Ġ=0ĊĊPoint3:Ġx3Ġy2:Ġ(23/(2Ġsqrt73
Rollout 86, Act: 3.345
Ġdx.ĊĊEvaluateĠterms:ĊĊAtĠpi/2:Ġ-Ġ(1/2)(pi/2)Ġcos
Top 16 Negative Activations
Rollout 39, Act: -2.362
21ĠtermsĠinĠtotal?ĠWait,Ġno.ĠWait,ĠletĠmeĠcheck.ĠIfĠtheĠfirstĠtermĠis
Rollout 22, Act: -2.379
e-23Ġkg?ĠWait,Ġno.ĠWait,Ġ0.002016
Rollout 20, Act: -2.386
ĠtheĠexponentĠofĠtheĠmultiplicativeĠgroupĠmoduloĠm.ĠButĠtheĠmultiplicativeĠgroupĠmoduloĠaĠprimeĠmĠisĠcyclic
Rollout 11, Act: -2.393
/bĠ+Ġa/cĠisĠtheĠreciprocalĠofĠq?ĠWait,ĠifĠweĠtakeĠreciprocalĠofĠeachĠtermĠinĠq
Rollout 40, Act: -2.400
/(1Ġ-ĠÏī)*(1Ġ+Ġi).ĠWait,Ġearlier:ĊĊz_{150}Ġ=
Rollout 83, Act: -2.401
Ġ1296.ĠButĠwait,Ġno.ĠWait:ĠifĠeachĠresidueĠisĠassignedĠtoĠexactlyĠone
Rollout 42, Act: -2.406
ĠadjacentĠcarbonsĠareĠnotĠonĠtheĠringsĠthemselves.ĠWait,Ġno.ĠLetĠmeĠthinkĠagain.ĠTheĠalcohol
Rollout 20, Act: -2.421
(mĠ-Ġ1)?ĠWait,Ġno,Ġwait:ĠTheĠexponentĠofĠaĠcyclicĠgroupĠisĠequalĠtoĠthe
Rollout 26, Act: -2.422
.71ĠkJ/molĊĊWaitĠaĠsecond.ĠButĠholdĠon.ĠTheĠenthalpyĠofĠformation
Rollout 80, Act: -2.440
0^26Ġgrams?ĠWait,ĠholdĠon:Ġ12.8Ġ*Ġ7.2
Rollout 86, Act: -2.444
ĠshouldĠintegrateĠtoĠtheĠsameĠvalueĠbecauseĠofĠsymmetry.ĠWait,ĠletĠmeĠverify.ĊĊButĠactually,ĠletĠme
Rollout 46, Act: -2.456
,ĠthatĠgivesĠtheĠaccelerationĠasĠtheĠgradient.ĠButĠwait,ĠinĠrotatingĠframes,ĠthereĠareĠadditionalĠterms,
Rollout 26, Act: -2.471
71ĠkJ/molĊĊWaitĠaĠsecond.ĠButĠholdĠon.ĠTheĠenthalpyĠofĠformationĠof
Rollout 40, Act: -2.476
Wait,ĠactuallyĠfactoringĠoutĠÏī^n,ĊĊWait,ĠletĠmeĠgroupĠtheĠterms:ĊĊFirstĠterm:
Rollout 69, Act: -2.536
0/9°,Ġetc.?ĠWait,Ġwait:Ġwait,ĠwhenĠi=0:Ġ30
Rollout 9, Act: -2.554
,ĠEMĠisĠtheĠtopĠsideĠofĠtheĠsquare.ĠWait,ĠholdĠon.ĠĊĊWait,ĠinĠtheĠsquare

DOWN_PROJ

Top 16 Positive Activations
Rollout 14, Act: 3.386
)Ġsqrt(AĠ)Ġ+K/A.ĊĊSetĠderivativeĠwithĠrespectĠtoĠAĠtoĠzero.ĊĊdh/dA=
Rollout 71, Act: 3.268
ĠtoĠsumsĠandĠproductsĠofĠitsĠroots.ĠLetĠmeĠrecallĠthemĠforĠaĠcubicĠequation.ĠIfĠtheĠrootsĠare
Rollout 52, Act: 3.215
ĠformulaĠforĠtheĠsumĠofĠconsecutiveĠnumbers.ĠLetĠmeĠremember.ĠIfĠyouĠhaveĠaĠsequenceĠstartingĠatĠaĠand
Rollout 14, Act: 3.205
ĠfixedĠ$J$ĠandĠ$K$.ĠIfĠIĠcanĠshowĠthatĠtheĠmaximumĠpossibleĠvalueĠofĠ$I^
Rollout 38, Act: 3.171
Ġsqrt=527ĠbecauseĠ527*527=?ĊĊWell,Ġ50
Rollout 51, Act: 3.123
ĠÏĢ.ĠHmm,Ġokay.ĠLetĠmeĠseeĠhowĠtoĠapproachĠthis.ĊĊFirstĠoff,ĠtheĠintegralĠinvolvesĠan
Rollout 99, Act: 3.113
Ġ10^0.334ĠâīĪĠ2.157),ĠsoĠ0
Rollout 19, Act: 2.999
2}Ġ\right)^2Ġ\).ĠLetĠmeĠrecallĠthatĠformula.ĠYes,ĠthatâĢĻsĠright:Ġ\(
Rollout 40, Act: 2.976
ĠOrĠperhapsĠusingĠcomplexĠnumbers?ĠHmm.ĠLetĠmeĠrecallĠthatĠrotationsĠcanĠbeĠrepresentedĠusingĠcomplexĠnumbersĠbyĠmultiplying
Rollout 14, Act: 2.962
ĠwithĠexponentsĠthatĠareĠconjugate.ĠLetĠmeĠrecallĠHölder'sĠinequality:ĠforĠ$p$Ġand
Rollout 21, Act: 2.913
ĠlikeĠsin(ÏĢĠf(x))Ġ=Ġ0ĠwouldĠforceĠf(x)ĠtoĠbeĠinteger,ĠbutĠthe
Rollout 66, Act: 2.897
ĠinĠtwoĠways:Ġ(leg1Ġ*Ġleg2)/2,ĠorĠ(hypotenuseĠ*Ġaltitude)/
Rollout 22, Act: 2.885
,333Ġ*2,869âīĪ18,157,Ġ6,
Rollout 56, Act: 2.860
ĠaĠright-angledĠtriangleĠisĠ(baseĠ*Ġheight)/2.ĠHere,ĠtheĠlegsĠareĠA0A
Rollout 87, Act: 2.857
ĠthisĠkernelâĢĶmaybeĠtabulatedĠsomewhere.ĠLetĠmeĠrecall.ĠInĠfact,ĠinĠsomeĠtextbooks,ĠlikeĠGriffith
Rollout 82, Act: 2.850
ĠalgorithmĠforĠaĠmixed-radixĠsystem?ĠLetĠmeĠrecallĠhowĠHuffmanĠcodingĠworks.ĠForĠaĠbinaryĠcode,
Top 16 Negative Activations
Rollout 22, Act: -2.519
ĠwhichĠoneĠtheyĠwant.ĊĊWait,ĠtheĠquestionĠdoesĠsayĠ"meanĠmolecularĠspeed,"ĠsoĠthatĠwouldĠbeĠthe
Rollout 31, Act: -2.529
Ġ-1)Ġ=0.ĊĊSoĠtheĠcubicĠfactorsĠasĠ(xĠ+1)(x^2Ġ-x
Rollout 75, Act: -2.532
ĠandĠthenĠcombiningĠthem,ĠbutĠno,ĠtheĠproblemĠstatesĠthatĠtheĠsumĠofĠtheĠthreeĠsquaresĠcanĠbeĠexpressed
Rollout 49, Act: -2.539
ĠsubspaceĠisĠÏĥ-compact.ĠSoĠevenĠsubspacesĠthatĠareĠnotĠclosedĠorĠnotĠnice?ĠForĠexample
Rollout 76, Act: -2.541
-3,Ġcols1-2.ĊĊCombined,ĠrowsĠ1-2:ĠcolsĠ1-3
Rollout 59, Act: -2.548
Ġ(3/28)^2?ĊĊWait,ĠbutĠwait,ĠsoĠoriginallyĠeachĠofĠtheirĠtrialsĠisĠindependent
Rollout 8, Act: -2.549
Ġadjacent).ĠButĠtheĠwayĠtheĠproblemĠisĠphrased:Ġ"allĠtheĠredĠcardsĠlaidĠoutĠareĠadjacent
Rollout 79, Act: -2.560
,ĠN).ĠSoĠperhapsĠ"in"ĠisĠjustĠpartĠofĠtheĠclueĠorĠusedĠdifferently.ĊĊAlternatively,Ġmaybe
Rollout 84, Act: -2.569
ĠtwiceĠtheĠoriginalĠpurchasedĠamount?ĠHmm,ĠtheĠproblemĠsaysĠ"twiceĠasĠmuch".ĠWhenĠweĠsayĠsomeone
Rollout 79, Act: -2.586
Ġetc.Ġ"Noticed"ĠmightĠbeĠtheĠanagramĠindicator.Ġ"MinorĠchanges"ĠcouldĠbeĠtheĠletters
Rollout 46, Act: -2.586
ĠAlternatively,ĠifĠtheĠCoriolisĠtermĠisĠpartĠofĠtheĠpotential,ĠbutĠit'sĠnot.ĠTheĠCor
Rollout 43, Act: -2.590
ĠtoĠ|1⣩.ĠSoĠinĠtheĠinputĠstate,ĠweĠhave:ĊĊα|00⣩ĠbecomesĠα
Rollout 42, Act: -2.601
cyclopropane-derived)ĠandĠaĠfour-memberedĠ(cyclobutane-derived)Ġring
Rollout 54, Act: -2.765
.Ġ4-methylcyclohexan-1-olateĠ(bulkyĠalkoxide,Ġdespite
Rollout 79, Act: -2.913
ĠN).ĠSoĠperhapsĠ"in"ĠisĠjustĠpartĠofĠtheĠclueĠorĠusedĠdifferently.ĊĊAlternatively,ĠmaybeĠonly
Rollout 36, Act: -3.018
Ġ3s^4ĊĊTotalĠs2Ġterms:Ġ1,728Ġ+37,

Layer 31

GATE_PROJ

Top 16 Positive Activations
Rollout 18, Act: 4.487
Ġ5).ĠButĠletĠmeĠdouble-checkĠonceĠagainĠinĠcaseĠofĠmistakes.ĊĊFirst,ĠcheckĠthatĠH_{
Rollout 94, Act: 4.478
verifyĠinĠcaseĠIĠmadeĠaĠmistake:ĊĊWait,ĠinĠstateĠ(d):ĠcoefficientsĠareĠa=1,
Rollout 72, Act: 4.385
ĠevenĠpositiveĠintegers.ĠWait,ĠbutĠlet'sĠcheckĠinĠtheĠproblemĠstatement.ĠAreĠweĠsupposedĠtoĠanswerĠthat
Rollout 94, Act: 4.303
ĠisĠcorrect.ĊĊWait,ĠletĠmeĠcross-verifyĠinĠcaseĠIĠmadeĠaĠmistake:ĊĊWait,ĠinĠstate
Rollout 45, Act: 4.213
ĠxĠ+ĠyĠ=Ġ0,ĠbutĠlet'sĠsee.ĊĊAlternatively,ĠthinkĠaboutĠsettingĠxĠorĠyĠto
Rollout 81, Act: 4.211
22/7ĊĊWaitĠaĠsecond,ĠnoteĠinĠthisĠapproachĠweĠgetĠ20h/(h
Rollout 39, Act: 4.176
Ġ1ĠandĠthenĠaĠ0.ĠWait,ĠtheĠnumberĠofĠonesĠcorrespondsĠtoĠtheĠexponentĠminusĠ1
Rollout 99, Act: 4.175
ĠtoĠseeĠifĠthereĠareĠerrors.ĊĊWait,ĠperhapsĠinĠtheĠsignĠforĠtheĠdenominator.ĠLetĠmeĠgoĠback
Rollout 45, Act: 4.174
4x2Ġ-4.ĊĊButĠwait,ĠbutĠinĠtheĠfunctionalĠequation,Ġaxy(xĠ+Ġy)+
Rollout 96, Act: 4.168
90ĊĊSolvingĠforĠvariables:ĊĊAlternatively,ĠsolveĠforĠoneĠvariableĠfromĠoneĠequationĠandĠsubstitute.ĊĊFrom
Rollout 42, Act: 4.152
ĠbitĠcomplicated.ĠLet'sĠlookĠatĠtheĠanswerĠchoicesĠandĠseeĠifĠthatĠcanĠhelp.ĊĊTheĠanswerĠoptionsĠare
Rollout 73, Act: 4.110
Ġcomplicated.ĠLetĠmeĠtryĠcalculatingĠforĠsmallĠn.ĊĊWhenĠn=1:ĠsumĠisĠ1*2
Rollout 94, Act: 4.109
ĠmeĠcross-verifyĠinĠcaseĠIĠmadeĠaĠmistake:ĊĊWait,ĠinĠstateĠ(d):ĠcoefficientsĠareĠa
Rollout 13, Act: 4.106
ĠthanĠ1.5625.ĊĊButĠtheĠexactĠcalculationĠgivesĠaroundĠ1.573
Rollout 73, Act: 4.095
16398.ĊĊButĠwait,ĠwhenĠIĠcomputeĠ16398ĠdividedĠbyĠ
Rollout 45, Act: 4.086
Ġb,ĠcĠbyĠpluggingĠknownĠvalues.ĊĊLetĠmeĠtryĠpluggingĠxĠ=Ġ1,ĠyĠ=
Top 16 Negative Activations
Rollout 83, Act: -1.570
8}ĊĊGivenĠtheĠconstraintĠthatĠtheĠsumĠofĠeveryĠthreeĠconsecutiveĠdigitsĠmustĠbeĠdivisibleĠbyĠ3,Ġwe
Rollout 49, Act: -1.587
Ġ\(ĠXĠ\)ĠisĠ\(\sigma\)-compact,ĠthenĠ\(ĠXĠ\)ĠisĠcountable,
Rollout 10, Act: -1.594
ĠunderstandĠtheĠgameĠfirst.ĊĊPlayersĠtakeĠturnsĠremovingĠaĠsquareĠnumberĠofĠpebblesĠ(1,Ġ4
Rollout 49, Act: -1.594
ffĠspaceĠĠ$X$ĠĠisĠĠ$\sigma$Ġ-compact,ĠthenĠĠ$X$Ġ
Rollout 26, Act: -1.629
3ĊĠĠĠ\]ĊĠĠĠUsingĠRĠ=Ġ8.3ĠJ/K/molĠ=Ġ0.0
Rollout 49, Act: -1.639
ĠspaceĠ\(ĠXĠ\)ĠisĠ\(\sigma\)-compact,ĠthenĠ\(ĠXĠ\)ĠisĠcountable
Rollout 10, Act: -1.642
ĠanalyzeĠtheĠgameĠwhereĠplayersĠalternatelyĠremoveĠaĠsquareĠnumberĠofĠpebbles.ĠTheĠkeyĠisĠtoĠidentify
Rollout 80, Act: -1.652
{Volume}Ġ=Ġ\frac{4}{3}Ġ\piĠr^3Ġ=Ġ\frac{4
Rollout 6, Act: -1.660
}$ĠĠwithĠaĠconstraintĠthatĠthereĠshouldĠexistsĠaĠpolynomialĠtimeĠalgorithmĠthatĠcanĠ**compute**ĠtheĠangelĠstring
Rollout 10, Act: -1.701
ĠtheĠGame**:ĊĠĠĠ-ĠPlayersĠtakeĠturnsĠremovingĠsquareĠnumbersĠ(1,Ġ4,Ġ9,
Rollout 10, Act: -1.756
ĠtheĠgameĠwhereĠplayersĠalternatelyĠremoveĠaĠsquareĠnumberĠofĠpebbles.ĠTheĠkeyĠisĠtoĠidentifyĠlosing
Rollout 10, Act: -1.783
bblesĠwhereĠĠ$x$ĠĠisĠtheĠsquareĠofĠanyĠpositiveĠinteger.ĠTheĠplayerĠwhoĠisĠunable
Rollout 57, Act: -1.796
Ġ1Ġ\).ĠBothĠsetsĠareĠcountableĠandĠthusĠhaveĠmeasureĠzero.Ċ2.Ġ**ClosureĠinĠDensity
Rollout 10, Act: -1.973
ĠweĠanalyzeĠtheĠgameĠwhereĠplayersĠalternatelyĠremoveĠaĠsquareĠnumberĠofĠpebbles.ĠTheĠkeyĠisĠto
Rollout 6, Act: -1.992
{P}_{angel}$ĠĠifĠthereĠexistsĠaĠpolynomialĠĠ$pĠ:Ġ\mathbb{N}
Rollout 31, Act: -1.999
ĠrealĠnumbersĠ$x,Ġy$.ĠForĠanyĠpairĠofĠrealĠnumbersĠ$a_{0},Ġa_{1

UP_PROJ

Top 16 Positive Activations
Rollout 56, Act: 3.587
3âĪļ3/4ĠâīĪĠ1.299.ĠOkay,ĠsoĠKĠmustĠbeĠat
Rollout 93, Act: 3.571
Ġm=4:9/8=1.125.ĠNotĠobvious.ĠMaybeĠtheĠclosed-form
Rollout 14, Act: 3.504
ĠtheĠmaximumĠpossibleĠratioĠisĠapproximatelyĠ2.25/8âīĪ0.28.ĠWait,
Rollout 14, Act: 3.503
ĠButĠthisĠseemsĠthatĠtheĠratioĠisĠ2.25/8=0.2812
Rollout 14, Act: 3.447
ĠbutĠtheĠmaximumĠpossibleĠratioĠisĠapproximatelyĠ2.25/8âīĪ0.28.ĠWait
Rollout 93, Act: 3.432
4:9/8=1.125.ĠNotĠobvious.ĠMaybeĠtheĠclosed-formĠisĠin
Rollout 14, Act: 3.426
ĠthisĠseemsĠthatĠtheĠratioĠisĠ2.25/8=0.28125
Rollout 14, Act: 3.410
)/(4/(k^3Ġ))=1/4.ĠWhichĠmatchesĠtheĠratioĠhere.ĠHmmm.ĊĊ
Rollout 56, Act: 3.367
Ġ3âĪļ3Ġ/4ĠâīĪ1.299),ĠasĠcomputedĠabove.ĠSoĠtheĠarea
Rollout 93, Act: 3.285
/32âīĪ1.03125,Ġm=4:9/8=1
Rollout 44, Act: 3.225
Ġ2.ĠTheĠreversibleĠworkĠwasĠ1.386.ĠSoĠyes,Ġhigher.ĠButĠin
Rollout 38, Act: 3.204
.428x.ĠSoĠLâīĪ1.857xĠ<ĠWâīĪ3.4
Rollout 14, Act: 3.185
5/8=0.28125.ĠWhichĠindicatesĠthatĠequalityĠisĠachievedĠinĠotherĠcases
Rollout 81, Act: 3.140
11=46ĠandĠ4/11.ĊĊButĠtheĠquestionĠstatesĠ"WhatĠisĠtheĠarea
Rollout 56, Act: 3.122
3âĪļ3Ġ/4ĠâīĪĠ1.299.ĠThenĠweĠneedĠtoĠcomputeĠtheĠareas
Rollout 93, Act: 3.104
1/8=9/8=1.125.ĊĊHmm,ĠsoĠforĠm=4
Top 16 Negative Activations
Rollout 63, Act: -2.217
Ġ\)ĠisĠzero.ĠOtherwise,ĠwhenĠbothĠ\(ĠpĠ\)ĠandĠ\(ĠqĠ\)ĠareĠpositive,
Rollout 38, Act: -2.226
.2Ġsqrt(x2Ġ+ĠW2)ĊĊLetĠmeĠexpressĠ6.2ĠasĠ31/
Rollout 9, Act: -2.228
25.ĊĊPerfect.ĊĊ**FinalĠAnswer**Ċ\boxed{25}ĊĊ<|im_start|>answerĊSquareĠ\(A
Rollout 93, Act: -2.234
014}}}ĠsinceĠtheĠclearestĠexpression.ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{1
Rollout 40, Act: -2.239
1Ġ-omega),ĠconjugateĠoverĠdenominator.ĊĊWait,Ġactually,ĠweĠhaveĠ1/(1Ġ-omega
Rollout 96, Act: -2.243
Ġcosθ)ĊĊButĠIĠneedĠtoĠverify.ĊĊWait,Ġlet'sĠconsiderĠthat.ĠLet'sĠsayĠweĠhave
Rollout 80, Act: -2.249
64ĠÃĹĠ10^26Ġgrams?ĠWait,ĠholdĠon:Ġ12.8
Rollout 52, Act: -2.251
013}.ĊĊ**FinalĠAnswer**Ċ\boxed{-2013}ĊĊ<|im_start|>answerĊTo
Rollout 80, Act: -2.334
ĠkmĠ=Ġ10^5Ġcm).ĠWait,Ġ1ĠkmĠ=Ġ1000Ġmeters
Rollout 96, Act: -2.350
ĠnewĠx-axisĠwithĠtheĠoriginalĠx-axis.ĊĊWait,Ġactually,ĠwhenĠconvertingĠaĠpointĠfromĠoriginalĠsystemĠto
Rollout 64, Act: -2.374
Ġ=zĠ+1Ġ+ĠyĊĊWait,ĠbutĠIĠhadĠpreviousĠsubstitutionĠz=xĠ-1.ĠSoĠx
Rollout 63, Act: -2.396
ĠOtherwise,ĠwhenĠbothĠ\(ĠpĠ\)ĠandĠ\(ĠqĠ\)ĠareĠpositive,ĠtheĠfunctionĠsatisfiesĠtheĠequation
Rollout 30, Act: -2.397
2,ĠwhichĠisĠâī¥2?ĠWait,Ġhmm,ĠtheĠrecursionĠforĠa_{n+1}Ġis
Rollout 33, Act: -2.430
})!Ġ*Ġ(6Ġ-Ġkm)!.ĊĊWait,ĠletĠmeĠcheckĠwithĠaĠconcreteĠexample.ĠTakeĠA
Rollout 91, Act: -2.450
cĠ-Ġb)/2,Ġ0).ĠWait,ĠbutĠactually,ĠifĠweĠhaveĠBCĠofĠlength,
Rollout 56, Act: -2.537
4-2}=ĠA4ĠA2.ĠWait,ĠforĠn=4,Ġn-2=2

DOWN_PROJ

Top 16 Positive Activations
Rollout 96, Act: 3.316
Ġ=132Ġ+12yĠ=>x=(132Ġ+12y)/5ĊĊ
Rollout 7, Act: 3.306
32x=8N/3Ġ=>Ġx=(8N)/(3*32)=N/
Rollout 96, Act: 3.090
Ġ=90Ġ-12xĊĊThereforeĠy=(90Ġ-12x)/5ĊĊPl
Rollout 96, Act: 2.916
=Ġ-1134ĊĊThus,Ġy=Ġ-1134/169.ĊĊ
Rollout 95, Act: 2.869
Ġ(y/H)^2=Ġk/NĠ=>ĠyĠ=ĠH*sqrt(k/N).ĊĊĠHence,ĠtheĠheight
Rollout 65, Act: 2.831
a2Ġ=1995ĠcĠâĩĴĠc=Ġa2Ġ/1995.ĊĊSimilarly,
Rollout 18, Act: 2.811
5bĠ=Ġ-3Ġ-4cĠ=>ĠbĠ=Ġ(-3Ġ-4c)/5ĊĊThisĠneeds
Rollout 49, Act: 2.791
compact,ĠweĠgetĠaĠcontradiction.ĊĊWait,ĠbutĠmaybeĠsuchĠaĠspaceĠXĠmustĠbeĠhereditarilyĠLind
Rollout 5, Act: 2.782
(xĠ+Ġ175)HĠ[1Ġ-Ġ125/(75Ġ+Ġx)]
Rollout 89, Act: 2.762
6ĠTĠ=Ġ100.ĠSoĠTĠ=Ġ100Ġ/0.6ĠâīĪ
Rollout 10, Act: 2.731
ĠconsideringĠthatĠbetweenĠconsecutiveĠsquaresĠ(k2ĠandĠ(k+1)2),ĠtheĠpossibleĠmovesĠthatĠcan
Rollout 82, Act: 2.692
ĠmustĠbeĠsuchĠthatĠgroupsĠwithĠhigherĠtotalĠprobabilityĠhaveĠshorterĠaverageĠbinaryĠcodewordĠlengths.ĊĊButĠactually,
Rollout 82, Act: 2.688
ĠbinaryĠsubtree.ĠSo,ĠassigningĠmoreĠprobableĠsymbolsĠtoĠshorterĠcodesĠinĠtheirĠrespectiveĠsubtrees.ĊĊWait,Ġthe
Rollout 18, Act: 2.679
:ĊĊ5ĠyĠ=xĠcĠ+4Ġ=>ĠyĠ=Ġ(xĠcĠ+4)/5ĊĊSetĠequal
Rollout 11, Act: 2.671
Ġ+Ġy2Ġ)=0ĠâĨĴĠyĠ[1Ġ-1/(x2Ġ+Ġy2Ġ)Ġ]=
Rollout 10, Act: 2.665
ĠWĠifĠthereĠexistsĠaĠsquareĠnumberĠsĠ<=nĠsuchĠthatĠnĠ-ĠsĠisĠL.ĊĊnĠisĠL
Top 16 Negative Activations
Rollout 4, Act: -2.775
ĠapproximatelyĠ0.36%,ĠconsistentĠwithĠtheĠpreviousĠresult.ĊĊTherefore,ĠcombiningĠthisĠwithĠtheĠ2
Rollout 91, Act: -2.778
2?ĊĊButĠwait,Ġwait.ĠSoĠaccordingĠtoĠmyĠcomputationsĠinĠthisĠcoordinateĠsystem,ĠtheĠtwoĠtermsĠ(
Rollout 3, Act: -2.791
ĠinĠkineticĠtheory,ĠtheyĠareĠdifferent.ĠLetĠmeĠmakeĠsure.ĊĊAccordingĠtoĠMaxwell-BoltzmannĠdistribution
Rollout 36, Act: -2.798
ĠtoĠattemptĠfactoringĠtheĠquarticĠequation.ĊĊSoĠcurrentĠequationĠis:ĊĊs^4Ġ-30,
Rollout 30, Act: -2.807
n+1}Ġb_nĠ-1.ĠSoĠourĠnumeratorĠisĊĊa_{n+1}Ġb
Rollout 81, Act: -2.811
.ĠNow,ĠisĠthereĠanyĠpossibleĠmiscalculationĠhere?ĊĊAlternatively,ĠletĠmeĠthinkĠofĠanotherĠapproach
Rollout 73, Act: -2.814
ĠfromĠk=1ĠtoĠn,ĠsoĠsubtractĠtheĠk=0ĠtermĠ(whichĠisĠ0),Ġso
Rollout 20, Act: -2.843
43,Ġp^{kn}Ġâī¡Ġ1ĠmodĠ43.ĠSimilarly,ĠforĠmodulusĠ
Rollout 80, Act: -2.867
ĠtoĠcalculateĠtheĠtotalĠnumberĠofĠneutrinosĠpassingĠthroughĠtheĠcoreĠperĠsecond.ĠButĠwait,ĠtheĠfluxĠdensity
Rollout 30, Act: -2.904
n+1Ġ}ĠisĠequalĠto:ĊĊFirstĠtermĠwhenĠn=1:Ġ2013Ġ/
Rollout 15, Act: -2.918
ĠnotĠcycling.ĠTherefore,ĠmyĠmistakeĠwasĠintroducingĠthatĠpreviously.ĠSoĠmaybeĠn=3Ġisn'tĠcyclic.ĊĊ
Rollout 43, Act: -2.952
>.ĊĊIĠneedĠtoĠbeĠcarefulĠhere.ĠSoĠinĠmyĠcalculation,ĠcoefficientsĠmatter.ĠButĠletĠmeĠredoĠonce
Rollout 30, Act: -3.064
'sĠadjustĠindicesĠforĠtheseĠsums.ĠLetĠkĠ=ĠnĠ-1ĠforĠtheĠfirstĠsumĠandĠmĠ=Ġn
Rollout 97, Act: -3.101
Ġ~2.3.ĊĊButĠthenĠwhyĠwasĠtheĠpreviousĠanswerĠapproxĠ4.6?ĊĊBecause,Ġin
Rollout 84, Act: -3.234
ĠtwiceĠtheĠfraction,ĠorĠtwiceĠtheĠvolume.ĊĊInĠmyĠinitialĠinterpretation,Ġit'sĠtwiceĠtheĠvolume.ĠThat
Rollout 69, Act: -3.436
).ĊĊSo,ĠforĠS(theta),ĠfirstĠtermĠisĠi=0:Ġsin2(theta)/cos(3

Layer 32

GATE_PROJ

Top 16 Positive Activations
Rollout 68, Act: 1.019
ĠquadraticĠequationĠhasĠtwoĠrealĠsolutions.ĠHowever,ĠweĠneedĠtoĠcheckĠifĠtheseĠsolutionsĠlieĠwithinĠtheĠdomainĠconstraints
Rollout 70, Act: 0.871
ĠweĠverifyĠ\(Ġ2Ġ\timesĠ315Ġ=Ġ630Ġ\).ĊĊThus,Ġthe
Rollout 49, Act: 0.853
rizable,ĠconsiderĠaĠBernsteinĠset.ĠAĠBernsteinĠsetĠisĠuncountable,ĠmeetsĠeveryĠperfectĠset,
Rollout 49, Act: 0.846
?ĊĊWait,ĠmaybeĠtheĠBernsteinĠset.ĠAĠBernsteinĠsetĠisĠaĠsubsetĠBĠofĠaĠPolishĠspaceĠsuchĠthat
Rollout 68, Act: 0.808
ĠtheĠquadraticĠequationĠhasĠtwoĠrealĠsolutions.ĠHowever,ĠweĠneedĠtoĠcheckĠifĠtheseĠsolutionsĠlieĠwithinĠtheĠdomain
Rollout 72, Act: 0.767
:Ġ\(Ġe^{2\piĠiĠk/5}Ġ\)ĠforĠk=1,2,
Rollout 93, Act: 0.764
ĠcorrespondsĠtoĠanglesĠthatĠareĠÏĢ/1007,Ġ2ÏĢ/1007,
Rollout 72, Act: 0.763
Ġn=5ĠareĠ1ĠandĠtheĠprimitiveĠ5thĠroots:Ġ\(Ġe^{2\piĠi
Rollout 68, Act: 0.753
Ġ+Ġ4Ġ=Ġ0Ċ\]ĊĊWeĠneedĠthisĠquadraticĠequationĠtoĠhaveĠexactlyĠoneĠrealĠsolutionĠin
Rollout 33, Act: 0.738
Ġ\(1Ġ\leqĠkĠ\leqĠ5\),ĠatĠleastĠoneĠofĠtheĠfirstĠ\(k
Rollout 83, Act: 0.732
agon,ĠsoĠeveryĠsetĠofĠthreeĠconsecutiveĠverticesĠwrapsĠaround.ĊĊFirst,ĠmaybeĠIĠshouldĠthinkĠaboutĠtheĠpossible
Rollout 6, Act: 0.724
Ġ\)ĠunlessĠ\(Ġ\textbf{NP}Ġ=Ġ\textbf{P}Ġ\).ĊĊ---ĊĊ**
Rollout 2, Act: 0.695
91Ġmod125:Ġ125*7=875.Ġ89
Rollout 90, Act: 0.682
:ĊĊ1,928ĠisĠlessĠthan2018.ĠTherefore,10,00
Rollout 93, Act: 0.681
}^{N-1}Ġe^{i2ÏĢkĠm/N}=NĠifĠmĠâī¡0
Rollout 72, Act: 0.670
=5.ĠTheĠrootsĠofĠunityĠforĠn=5ĠareĠ1ĠandĠtheĠprimitiveĠ5thĠroots
Top 16 Negative Activations
Rollout 68, Act: -3.683
ĠfromĠ-500ĠtoĠ500ĠisĠ500ĠnegativeĠintegersĠ(k=-1
Rollout 70, Act: -3.687
7^0.ĊĊButĠifĠweĠtakeĠthoseĠdigitsĠd_kĠd_{k-1}Ġ...Ġd_
Rollout 38, Act: -3.688
ĠandĠWĠsouth,ĠsoĠtheĠdistanceĠisĠsqrt(x2Ġ+ĠW2).ĠTherefore,ĠhisĠtimeĠisĠsqrt
Rollout 83, Act: -3.693
ĠsameĠresidue.ĠRemember,ĠresidueĠcounts:Ġ0,1,2ĠeachĠneedsĠtoĠbeĠassignedĠexactlyĠthree
Rollout 61, Act: -3.696
ĠofĠm's:ĠforĠd,Ġit'sĠmĠfromĠ0ĠtoĠ39Ġ-d,Ġhence
Rollout 35, Act: -3.709
0.ĊĊLetâĢĻsĠtestĠDĠ=1:Ġb2Ġ+7bĠ-1=0.ĠDiscriminant
Rollout 22, Act: -3.724
2Ġ*Ġ1.6605ĠisĠaboutĠ3.321.ĠThenĠ0
Rollout 71, Act: -3.748
ĠmultipleĠpossibleĠm,ĠbutĠweĠjustĠneedĠtheĠsumĠ|a|Ġ+Ġ|b|Ġ+Ġ|c|
Rollout 10, Act: -3.774
IfĠnĠâī¡0Ġmod4:Ġsubtract0Ġor1.ĊĊĠsubtractĠsâī¡0Ġor1Ġmod4
Rollout 31, Act: -3.779
1.ĠIndeed,ĠthereĠwouldĠbeĠcoefficientsĠc_nĠandĠd_nĠsuchĠthatĠa_nĠ=Ġc_nĠa
Rollout 4, Act: -3.795
ĠsignificantĠfigures,ĠsoĠ0.88%,ĠorĠmaybeĠtheĠanswerĠisĠexactĠatĠ0.8
Rollout 0, Act: -3.835
1.ĊĊSoĠprimesĠ2:Ġ18,Ġ3:Ġ8,Ġ5:4
Rollout 91, Act: -3.836
ĠADĠisĠtan(45)Ġ=Ġ1ĠorĠtan(135)Ġ=Ġ-1.
Rollout 53, Act: -3.862
3))Ġ)ĠRationallze:ĠthisĠisĠorĠweĠcanĠleaveĠasĠitĠis.ĠMaybeĠweĠcan
Rollout 72, Act: -3.927
.ĠThenĠweĠwantĠtoĠsolveĠforĠexponentsĠi,Ġj,ĠkĠsuchĠthatĠÎ^iĠ+
Rollout 26, Act: -4.244
JĠ=Ġ296.0ĠkJĠbutĠonlyĠthreeĠsigĠfigs.ĠSoĠ29

UP_PROJ

Top 16 Positive Activations
Rollout 80, Act: 2.663
ĠEarth'sĠsurfaceĠisĠ10^11ĠneutrinosĠperĠsecondĠperĠsquareĠcentimeter.ĠĊĊThe
Rollout 80, Act: 2.656
Ġfirst,ĠweĠneedĠtoĠcalculateĠtheĠtotalĠnumberĠofĠneutrinosĠpassingĠthroughĠtheĠcoreĠperĠsecond.ĠButĠwait
Rollout 80, Act: 2.416
Ġis),ĠthenĠtheĠneutrinosĠareĠcomingĠinĠparallelĠrays.ĠTheĠnumberĠpassingĠthroughĠtheĠinnerĠcore'sĠcross
Rollout 80, Act: 2.378
ĠneutrinoĠflux:Ġ10^11Ġneutrinos/s/cm2.ĠHowĠmanyĠneutrinosĠhit
Rollout 74, Act: 2.371
LetâĢĻsĠdefine:ĊĊ-ĠH_e:ĠEarth'sĠhorizontalĠmagneticĠfieldĠtowardsĠnorth.ĊĊ-ĠH_ship_N:
Rollout 80, Act: 2.354
Ġsurface.ĊĊTherefore,ĠtoĠcalculateĠtheĠtotalĠnumberĠofĠneutrinosĠpassingĠthroughĠtheĠEarthĠperĠsecond,ĠweĠcould
Rollout 74, Act: 2.351
ĠdownĠtheĠcomponents.ĊĊAssumeĠthatĠtheĠEarth'sĠhorizontalĠmagneticĠfieldĠisĠH_e.ĠTheĠship'sĠmagnetic
Rollout 74, Act: 2.309
Ġcomponents.ĊĊAssumeĠthatĠtheĠEarth'sĠhorizontalĠmagneticĠfieldĠisĠH_e.ĠTheĠship'sĠmagneticĠfieldsĠare
Rollout 74, Act: 2.284
ĠTheĠcompassĠneedleĠpointsĠinĠtheĠdirectionĠofĠtheĠresultantĠhorizontalĠmagneticĠfield.ĠTheĠEarth'sĠfieldĠisĠH_e
Rollout 80, Act: 2.280
,ĠweĠneedĠtoĠcalculateĠtheĠtotalĠnumberĠofĠneutrinosĠpassingĠthroughĠtheĠcoreĠperĠsecond.ĠButĠwait,
Rollout 80, Act: 2.266
ĠEarth'sĠinnerĠcore",ĠwhichĠprobablyĠmeansĠallĠtheĠneutrinosĠthatĠhitĠtheĠEarth'sĠsurfaceĠalsoĠareĠabsorbed
Rollout 80, Act: 2.250
ĠdeterminesĠtheĠtotalĠpower.ĊĊTherefore,ĠtheĠnumberĠofĠneutrinosĠabsorbedĠperĠsecondĠwouldĠbeĠtheĠfluxĠ(1
Rollout 74, Act: 2.232
ĠH_eĠandĠH_sĠ=ĠK_sĠ*ĠH_e,ĠthenĠtanδθĠ=Ġ[K_bĠH
Rollout 80, Act: 2.220
11Ġneutrinos/s/cm2.ĠHowĠmanyĠneutrinosĠhitĠtheĠcore?ĠAssumingĠtheĠcoreĠisĠa
Rollout 80, Act: 2.184
ĠtheĠcenter,ĠbutĠtheĠcross-sectionalĠareaĠthatĠtheĠneutrinosĠpassĠthroughĠwouldĠbeĠtheĠcross-sectionĠofĠthe
Rollout 80, Act: 2.177
Ġsurface,ĠtheĠfluxĠisĠ1e11Ġneutrinos/s/cm2.ĠBecauseĠneutrinosĠtravelĠin
Top 16 Negative Activations
Rollout 73, Act: -2.363
$ĠisĠdividedĠbyĠ$1000$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI
Rollout 66, Act: -2.364
IfĠmĠandĠnĠareĠcoprimeĠandĠoppositeĠparity,ĠthenĠletĠmeĠconsider:ĊĊgcd(Ġm2
Rollout 72, Act: -2.364
ĠsupposeĠthatĠoneĠofĠthemĠisĠ1Ġ(i=0).ĠThenĠtheĠequationĠwouldĠbeĠ1Ġ+
Rollout 11, Act: -2.366
ateĠofĠvariables.ĠLetâĢĻsĠsetĠxĠ=Ġ\overline{a},ĠyĠ=Ġ\overline{
Rollout 41, Act: -2.368
ĠbothĠdigitsĠareĠnon-zero.ĠThen,ĠweĠcanĠarrangeĠthoseĠtwoĠdigitsĠfreely,ĠbutĠensuringĠnoĠthreeĠof
Rollout 66, Act: -2.372
2Ġ+Ġn2ĊĊSo,Ġlet'sĠcomputeĠallĠterms:ĊĊFirst,ĠaĠ=Ġm2Ġ-Ġn
Rollout 66, Act: -2.386
Ġhere.ĠTherefore,ĠmĠandĠnĠmustĠhaveĠoppositeĠparity.ĠHence,ĠtheĠparameterizationĠrequiresĠthatĠinĠa
Rollout 11, Act: -2.395
Ġonly.ĊĊAlternatively,ĠmaybeĠtakeĠconjugateĠofĠvariables.ĠLetâĢĻsĠsetĠxĠ=Ġ\overline{a
Rollout 66, Act: -2.406
ence,ĠmĠandĠnĠhaveĠtoĠbeĠofĠoppositeĠparity.ĠIfĠm=13ĠisĠodd,Ġn
Rollout 71, Act: -2.409
ĠsignsĠaccordingly,Ġmaybe.ĠForĠexample,ĠswappingĠeĠandĠd,ĠbutĠsinceĠcoefficientĠisĠ3,Ġif
Rollout 11, Act: -2.437
'sĠdenoteĠxĠ=Ġa,ĠyĠ=Ġb,ĠzĠ=Ġc.ĠThen,ĠgivenĠxyzĠ=Ġ1
Rollout 71, Act: -2.441
ĠhaveĠallĠeven.ĊĊSimilarly,ĠtwoĠevenĠandĠoneĠodd:ĠLetĠtwoĠbeĠeven,ĠoneĠodd.ĠThen
Rollout 61, Act: -2.457
2^kĠ+Ġ2^mĠâī¡Ġ0ĠmodĠ9.ĠLetâĢĻsĠwriteĠthisĠas
Rollout 11, Act: -2.496
:ĊĊGivenĠabc=1,ĠthenĠtakingĠreciprocals,Ġ(1/a)(1/b)(1/c
Rollout 66, Act: -2.532
ĠforĠsomeĠmĠ>ĠnĠcoprimeĠintegersĠofĠoppositeĠparity.ĠOrĠmaybeĠtheyĠcouldĠbeĠinĠsomeĠother
Rollout 41, Act: -2.633
4=504.ĊĊCaseĠB:ĠOneĠofĠtheĠdigitsĠisĠzero.ĠHere,ĠthousandsĠdigitĠcan

DOWN_PROJ

Top 16 Positive Activations
Rollout 22, Act: 2.974
000,000=12,666ĊĊ0.0063
Rollout 9, Act: 2.902
celesĠtriangleĠwithĠbaseĠEM:ĠifĠit'sĠvertical,ĠthenĠtheĠbaseĠisĠvertical.ĠButĠtrianglesĠare
Rollout 42, Act: 2.878
ĠaĠcyclopropaneĠcarbon,ĠwhichĠoriginallyĠhadĠthreeĠbondsĠ(asĠpartĠofĠtheĠcyclopropane
Rollout 97, Act: 2.831
45:ĊĊ1.45=1.5Ġ-0.5Ġe^{-kt}ĠâĨĴĠ
Rollout 72, Act: 2.831
).ĠCos(36°)=Ġ(sqrt(5)Ġ+Ġ1)/4Ġ*Ġ2?
Rollout 90, Act: 2.800
14126.ĠMiddleĠdigitĠisĠtheĠthirdĠdigit,ĠwhichĠisĠ1.ĠRemovingĠgivesĠB
Rollout 42, Act: 2.758
lopropylĠgroup.ĠSoĠtheĠcentralĠcarbonĠhasĠthreeĠsubstituents:ĠoneĠhydroxylĠ(from
Rollout 42, Act: 2.719
ĠaĠfour-memberedĠringĠattachedĠviaĠoneĠofĠitsĠcarbonsĠtoĠtheĠcentralĠcarbon),ĠaĠcycloprop
Rollout 54, Act: 2.694
ĠinĠSN2,ĠaĠbulkyĠnucleophileĠ(likeĠtert-butoxide)ĠisĠlessĠnucleophilic
Rollout 9, Act: 2.681
ĠtriangleĠGEMĠwithĠbaseĠEM.ĠSoĠEMĠisĠoneĠofĠtheĠsidesĠofĠtheĠsquare,ĠtheĠbottomĠside
Rollout 72, Act: 2.681
72Ġdegrees)ĠisĠequalĠtoĠ(sqrt(5)Ġ-Ġ1)/4Ġ*Ġ2.
Rollout 9, Act: 2.673
Ġtriangle.ĠTheĠtriangleĠisĠisoscelesĠwithĠbaseĠEM,ĠsoĠtheĠlegsĠwouldĠbeĠGEĠandĠGM
Rollout 42, Act: 2.642
theĠspiroĠcarbon),ĠandĠtheĠdoubleĠbondĠatĠpositionĠ5.ĠWait,ĠtheĠnumberingĠwouldĠhaveĠthe
Rollout 42, Act: 2.639
ĠcarbocationĠisĠatĠtheĠcentralĠcarbonĠ(tertiaryĠmaybe?),ĠperhapsĠadjacentĠtoĠaĠthree-memberedĠring
Rollout 9, Act: 2.634
ĠapexĠofĠtheĠtriangleĠoppositeĠtheĠbaseĠEM,ĠwhichĠisĠtheĠverticalĠsideĠofĠtheĠsquare.ĠIfĠtheĠaltitude
Rollout 72, Act: 2.633
)ĠisĠknownĠtoĠbeĠ(1Ġ+Ġsqrt(5))/4ĠmultipliedĠbyĠ2.ĠPerhapsĠaĠbetter
Top 16 Negative Activations
Rollout 30, Act: -2.433
n+1}ĠtoĠa_{n-1}ĠandĠb_n.ĊĊSoĠtoĠexpressĠa_{m
Rollout 18, Act: -2.435
ĠareĠintegers,ĠthenĠaĠxĠ+ĠbĠyĠ=ĠsĠxĠ+ĠtĠyĠwhereĠsĠ=ĠaĠandĠt
Rollout 50, Act: -2.446
First,ĠrecallĠthatĠ\(\log(aĠ\cdotĠb)Ġ=Ġ\logĠaĠ+Ġ\logĠb\
Rollout 81, Act: -2.458
20h/(51Ġ-h2)Ġ=22/7ĠbutĠinĠthisĠapproach,Ġby
Rollout 81, Act: -2.460
ĠNotĠright-angled,ĠButĠweĠcanĠmakeĠdecompositionĠviaĠaltitude.ĊĊConsiderĠtriangleĠABCĠaltitudeĠhaĠ=hĠleading
Rollout 49, Act: -2.479
orffĠspaceĠisĠregular.ĠLet'sĠsee:ĠTakeĠaĠpointĠxĠandĠaĠclosedĠsetĠCĠnotĠcontaining
Rollout 56, Act: -2.492
ĠthatĠherĠpastureĠneverĠexceedsĠanĠareaĠofĠ$K$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġso
Rollout 18, Act: -2.494
4).ĠToĠcheckĠifĠHĠisĠZ2,ĠweĠcanĠcomputeĠtheĠdeterminantĠofĠtheĠmatrixĠformedĠbyĠtwo
Rollout 18, Act: -2.494
ĠofĠallĠintegerĠlinearĠcombinationsĠofĠ(0,Ġx)ĠandĠ(1,Ġy).ĠThatĠis,Ġevery
Rollout 18, Act: -2.548
_{xy}ĠisĠgeneratedĠbyĠ(0,Ġx)ĠandĠ(1,Ġy),ĠthenĠelementsĠofĠH
Rollout 38, Act: -2.562
.ĠSo,ĠweĠhaveĠAna,ĠBob,ĠandĠCaoĠbikingĠatĠdifferentĠspeeds:Ġ8.6
Rollout 1, Act: -2.573
ĠlĠisĠnon-emptyĠifĠandĠonlyĠifĠr1Ġ+Ġr2Ġ>=ĠlĠandĠ|r1Ġ-Ġr
Rollout 18, Act: -2.616
Ġ2.ĠTherefore,ĠHĠisĠisomorphicĠtoĠZ2ĠifĠtheĠgeneratorsĠareĠlinearlyĠindependentĠoverĠZ
Rollout 11, Act: -2.629
ĠHmm,ĠcomplexĠnumbersĠwithĠproductĠ1.ĠSoĠabcĠ=Ġ1.ĠNoneĠofĠthemĠareĠrealĠor
Rollout 56, Act: -2.633
ĠgoĠtoĠinfinity.ĠTheĠquestionĠisĠessentiallyĠaskingĠforĠtheĠsumĠofĠtheĠseriesĠofĠareas,ĠifĠitĠconver
Rollout 58, Act: -2.659
ĠofĠkĠandĠsetĠupĠtheĠequationĠthatĠoneĠisĠ2018ĠtimesĠtheĠother.ĊĊLetĠme

Layer 33

GATE_PROJ

Top 16 Positive Activations
Rollout 62, Act: 4.912
17303.ĠLet'sĠsee:Ġ121*100=12
Rollout 81, Act: 4.879
).ĠCalculateĠsqrt(25909):ĠclosestĠsquareĠisĠ1612=25
Rollout 36, Act: 4.877
,662,592):ĊĊSinceĠ19,0002Ġ=Ġ3
Rollout 36, Act: 4.832
imatesĠtoĠ158.12ĠbecauseĠ1582=24,96
Rollout 96, Act: 4.797
·13:ĊĊ13*133Ġ=Ġ1729,ĠsoĠ17
Rollout 3, Act: 4.706
8,436).ĠLet'sĠsee,Ġ168^2Ġ=Ġ28,
Rollout 81, Act: 4.674
9584)Ġ=Ġlet'sĠcheck:Ġ172^2=2958
Rollout 81, Act: 4.651
=25,921.Ġ,SinceĠ1612=25,92
Rollout 16, Act: 4.621
80ĠdividedĠbyĠ1953:Ġ1953*4=781
Rollout 96, Act: 4.570
Ġby2197:ĊĊ2197*31=2197*30
Rollout 38, Act: 4.527
27*527=?ĊĊWell,Ġ5002=250,0
Rollout 3, Act: 4.509
8,408).ĠLet'sĠsee.ĊĊ168^2Ġ=Ġ(170Ġ-
Rollout 36, Act: 4.509
1,856=Ġlet'sĠcheck:ĊĊ10,000*10,0
Rollout 90, Act: 4.495
8073Ġ÷1009:Ġ1009*8=80
Rollout 53, Act: 4.454
1Ġ÷146:ĊĊ146*13=1898ĠâĨĴĠ1
Rollout 81, Act: 4.452
29584)Ġ=Ġlet'sĠcheck:Ġ172^2=295
Top 16 Negative Activations
Rollout 80, Act: -4.138
ĠpassĠthroughĠtheĠEarth,ĠandĠtheĠfluxĠisĠgivenĠatĠEarth'sĠsurface,ĠbutĠonlyĠthoseĠthatĠpassĠthrough
Rollout 27, Act: -4.163
ĠTĠ=Ġ(mĠv_rms2)/(3k).ĠThenĠonceĠweĠknowĠmĠ(massĠofĠBe
Rollout 25, Act: -4.164
e6ĊĊButĠa_earthĠ=ĠG*mĠ/Ġd2(t),ĠdependsĠonĠtime-varying
Rollout 97, Act: -4.168
ĠisĠwrittenĠas:ĊĊRateĠ=Ġ-Ġ(1/a)Ġd[A]/dtĠ=ĠkĠ[A]^n
Rollout 27, Act: -4.174
v_z2⣩Ġ=Ġ(1/2)kĠTĠâĨĴĠ⣨v_z2⣩Ġ=ĠkĠT
Rollout 27, Act: -4.175
Ġ=Ġ(2ĠÎĶE_rms)Ġ/Ġ(3k).ĠGetĠ2*5.54
Rollout 27, Act: -4.182
TĠâĩĴĠTĠ=Ġ2ÎĶE_rmsĠ/k.ĊĊOnceĠweĠhaveĠT,ĠbutĠsinceĠthe
Rollout 27, Act: -4.187
ĠleadĠtoĠTĠ=Ġ2ĠÎĶE_rmsĠ/Ġk.ĊĊThen,ĠonceĠTĠisĠknown,ĠT
Rollout 27, Act: -4.190
ĠTĠ=Ġ(2Ġ*ĠdeltaĠE_rms)Ġ/Ġk.ĊĊPluggingĠnumbers:ĊĊdeltaĠE_rms
Rollout 25, Act: -4.227
Ġt2Ġ=Ġ(1/2)*(ĠG*mĠ/d_min2Ġ)Ġ*Ġ(d_min/vĠ)
Rollout 80, Act: -4.354
ĠHowever,ĠifĠtheĠneutrinosĠareĠcomingĠuniformlyĠfromĠtheĠSun,ĠthenĠtheirĠpathsĠareĠparallel,ĠallĠcoming
Rollout 27, Act: -4.384
ÎĶE_rmsĠ=Ġ(1/2)kTĠâĩĴĠTĠ=Ġ2ÎĶE_rms
Rollout 27, Act: -4.405
ĠthusĠTĠ=ĠmĠV_rms2/(3k).ĠAndĠtheyĠcallĠthatĠTc.ĠPerhapsĠconfused
Rollout 12, Act: -4.428
1Ġ-k2)K(k)Ġ]Ġ/Ġ(k(1Ġ-k2))Ġ*Ġ(Ġ-
Rollout 80, Act: -4.452
.ĠHowever,ĠifĠtheĠneutrinosĠareĠcomingĠuniformlyĠfromĠtheĠSun,ĠthenĠtheirĠpathsĠareĠparallel,Ġall
Rollout 12, Act: -4.585
Ġ(1Ġ-k2)K(k)Ġ]Ġ/Ġ(k(1Ġ-k2))Ġ*Ġ(

UP_PROJ

Top 16 Positive Activations
Rollout 36, Act: 3.205
,760âīĪ2.016ĊĊThus,ĠapproximateĠsqrt(ÎĶ)=19,
Rollout 70, Act: 3.054
8Ġ=>d1Ġ<=2.666,ĠsoĠd1=0,1,2.ĊĊ
Rollout 23, Act: 2.932
91âīĪ0.05126.ĠSoĠthatĠ73.0422
Rollout 4, Act: 2.922
0036058ĠasĠaĠdecimal,ĠsoĠyes,Ġ0.3605
Rollout 68, Act: 2.894
6)/2âīĪ-0.008.ĠSoĠxâīĪ-0.008,
Rollout 23, Act: 2.868
2585âīĪ0.722.ĠSo,ĠapproximatelyĠ31.722
Rollout 3, Act: 2.777
)/337âīĪ0.047.ĠThus,ĠsqrtâīĪ168.5Ġ+
Rollout 17, Act: 2.733
*0.41=3.28,ĠsoĠtotalĠ11.28.ĊĊThus
Rollout 13, Act: 2.701
Ġ1.12550881,ĠthereforeĠ1.12550
Rollout 23, Act: 2.693
72.991âīĪ0.05126.ĠSoĠthatĠ73.
Rollout 22, Act: 2.677
=145,Ġ659.ĠRemainderĠisĠ145,458
Rollout 65, Act: 2.673
Ġk*cĠ->Ġk=1995/c,ĠandĠfromĠcĠ=Ġk*1995
Rollout 22, Act: 2.672
2*273=.ĠHmm,Ġ0.5*273=136.
Rollout 26, Act: 2.669
66=Ġ2.4734ĊĊThenĠmultiplyĠbyĠ-3:Ġ-3Ġ*Ġ
Rollout 36, Act: 2.651
400=78,192ĊĊSoĠ=>ĠdkâīĪ78,192
Rollout 36, Act: 2.635
;Ġ=>xâīĪ2.0105...ĊĊButĠlikely,Ġsqrt(36,86
Top 16 Negative Activations
Rollout 70, Act: -2.566
Ġ\(Ġd_kĠd_{k-1}Ġ\ldotsĠd_0Ġ\)ĠinĠbase-7
Rollout 46, Act: -2.568
Ġ+Ġ(1/2)Ïī2(x2Ġ+Ġy2)Ġ+ĠÏīĠ(xĠdy/d
Rollout 7, Act: -2.568
Ġ59ĠforĠnon-promotedĠparticipants.ĠĊĊ###ĠPartĠ(a)ĊĊ1.Ġ**OriginalĠequations:
Rollout 32, Act: -2.583
\omegaĠ\\Ġ1Ġ&Ġ0Ġ\end{pmatrix}\)Ċ2.Ġ\(\mathbf
Rollout 35, Act: -2.588
Ġ\)ĠandĠ\(ĠP(24)Ġ=Ġ17Ġ\),ĠweĠfind:Ċ\[ĊQ
Rollout 91, Act: -2.591
Ġ(tĠ-Ġ1,Ġt)\)ĠandĠ\(\overrightarrow{CB}Ġ=Ġ(-2,
Rollout 46, Act: -2.591
2Ġ-Ġ(Ïī2/2)(x2Ġ+Ġy2).ĠHmm,Ġright.ĠLetĠmeĠtry
Rollout 0, Act: -2.608
)ĠandĠ\(\gcd(a,Ġb)Ġ=Ġ1\),ĠeachĠprimeĠfactorĠofĠ20
Rollout 53, Act: -2.616
Ġ=Ġ2\)ĠimpliesĠ\(ADĠ=Ġ3Ġ\timesĠ2Ġ=Ġ6\).ĊĊ2.
Rollout 46, Act: -2.627
^2}{2}\left(x^{2}Ġ+Ġy^{2}\right)ĊĠĠĠ\]ĊĊ2
Rollout 6, Act: -2.635
ĠsetsĠ\(ĠS_1,ĠS_2,Ġ\ldots,ĠS_kĠ\)ĠbyĠtaggingĠeach
Rollout 25, Act: -2.635
ĠinĠaĠclosestĠapproachĠdistance.ĠInĠreality,Ġ2023ĠBU'sĠclosestĠapproachĠwasĠ3
Rollout 5, Act: -2.636
Ġ\)ĠofĠtheĠsegmentĠthatĠdividesĠtheĠtrapezoidĠintoĠtwoĠregionsĠofĠequalĠarea,ĠweĠuse
Rollout 5, Act: -2.679
00Ġ\).ĠTheĠmidlineĠofĠtheĠtrapezoid,ĠwhichĠconnectsĠtheĠmidpointsĠofĠthe
Rollout 97, Act: -2.707
ĠpressureĠ\(P_0Ġ=Ġ1Ġ\,Ġ\text{atm}\),ĠfinalĠpressureĠ\(P
Rollout 61, Act: -2.752
1Ġ\pmod{9}Ġ\).ĠThisĠsimplifiesĠtoĠfindingĠ\(ĠdĠ\)ĠsuchĠthatĠ\(

DOWN_PROJ

Top 16 Positive Activations
Rollout 35, Act: 3.358
Ġ(bĠ+7)*bĠmustĠbeĠaĠdivisorĠofĠ10.ĠTherefore,ĠweĠneedĠintegerĠvalues
Rollout 56, Act: 3.256
ĠstraightĠline.ĠSoĠA0ĠisĠbetweenĠA2ĠandĠA3.ĠSoĠtheĠlengthĠA3A0
Rollout 62, Act: 3.185
561÷29=88.31,Ġno.Ġ31:31
Rollout 67, Act: 2.963
)=Ġ-24/k.ĠTherefore,ĠkĠmustĠdivideĠ24.ĊĊSoĠkĠisĠanĠintegerĠthat
Rollout 65, Act: 2.909
2)*1995,ĠsoĠn2ĠdividesĠ1995*m2.ĠButĠsince
Rollout 71, Act: 2.902
96.ĠClose,ĠbutĠ85.81,Ġnot.Ġc=1:3*
Rollout 56, Act: 2.888
3ĠisĠaĠstraightĠline.ĠSoĠA0ĠisĠbetweenĠA2ĠandĠA3.ĠSoĠtheĠlengthĠA
Rollout 83, Act: 2.864
(9,Ġk)Ġ=Ġd,ĠsoĠdĠdividesĠ9.ĠThen,ĠrotationĠbyĠkĠwillĠsplit
Rollout 3, Act: 2.830
28,224.ĊĊ168.5^2=Ġ(168Ġ+Ġ
Rollout 67, Act: 2.813
ĠforĠpĠandĠqĠtoĠbeĠintegers,ĠkĠdividesĠintoĠ96ĠandĠ36.ĠTheĠgcd
Rollout 67, Act: 2.775
15d/2.ĊĊButĠaĠmustĠdivideĠintoĠ1440ĠasĠperĠequationĠ1
Rollout 67, Act: 2.775
SoĠ6a2Ġ+29ĠaĠdĠ-120Ġd2=0.ĊĊAnĠequation
Rollout 35, Act: 2.768
Ġ(bĠ+7)*bĠmustĠbeĠaĠdivisorĠofĠ10,ĠbecauseĠcĠhasĠtoĠbeĠinteger
Rollout 71, Act: 2.698
011/7=287.285...Ġ11:Ġ20
Rollout 32, Act: 2.695
0ÏīĊĊTerm3ĠhasĠ+670ĠandĠ-670Ïī2ĊĊTerm4Ġhas
Rollout 66, Act: 2.691
6b2Ġ-Ġ6bĠcĠ=Ġ0ĊĊHmm,ĠperhapsĠgroupĠterms:ĊĊa2Ġ-
Top 16 Negative Activations
Rollout 55, Act: -2.514
ĠinĠtheĠx-zĠplaneĠandĠaĠ1x1ĠreflectionĠmatrixĠinĠtheĠy-axis.ĠTheĠ2
Rollout 63, Act: -2.517
-Ġ\(Ġf(1,Ġ1)Ġ=Ġ1Ġ\)Ċ-Ġ\(Ġf(2,Ġ
Rollout 62, Act: -2.519
Ġ121,ĠwhichĠisĠ112.ĠThenĠm=12:Ġ14
Rollout 75, Act: -2.520
ĠaĠsumĠofĠtwoĠsquares.ĠForĠexample,ĠtheĠsquareĠofĠanĠoddĠintegerĠisĠ1ĠmodĠ4
Rollout 61, Act: -2.533
Ġ8ĠmodĠ9,ĠwhereĠdĠ=ĠkĠ-Ġm.ĠBecauseĠ-1ĠmodĠ9ĠisĠ
Rollout 55, Act: -2.539
2x2ĠrotationĠmatrixĠandĠtheĠ1x1ĠmatrixĠ[-1],ĠthenĠwhenĠweĠraiseĠAĠto
Rollout 15, Act: -2.560
'sĠtryĠn=3,ĠwhichĠisĠ2^2Ġ-1:Ġno,ĠsinceĠn=3
Rollout 62, Act: -2.651
11ĠyieldsĠ121=112.ĠButĠ(11k)^2Ġ-1
Rollout 55, Act: -2.665
Thus,ĠifĠAĠisĠaĠblockĠdiagonalĠmatrixĠwithĠblocks:ĠtheĠ2x2ĠrotationĠmatrixĠandĠthe
Rollout 95, Act: -2.695
-1}ĠandĠPkQkĠisĠtheĠaverageĠofĠtheĠlengthsĠofĠtheĠtwoĠsegmentsĠmultipliedĠbyĠthe
Rollout 17, Act: -2.712
ĠisĠunitless).ĠWait,ĠΨĠhasĠunitsĠofĠ1/m^(3/2),ĠsoĠ|
Rollout 5, Act: -2.819
ĠbasesĠthatĠdividesĠtheĠtrapezoidĠintoĠtwoĠregionsĠofĠequalĠarea,ĠitsĠlengthĠxĠshouldĠbeĠsuch
Rollout 3, Act: -2.883
ĠisĠaĠgasĠatĠstandardĠtemperatureĠandĠpressure,ĠwithĠboilingĠpointĠatĠ-61.7Ġ°C,
Rollout 15, Act: -2.894
1?ĠWait,Ġn=1ĠisĠ2^1Ġ-1.ĠLet'sĠtryĠn=3
Rollout 55, Act: -2.932
ĠinĠtheĠx-zĠplaneĠandĠaĠ1x1ĠmatrixĠ(-1)ĠactingĠonĠtheĠy-axis.
Rollout 72, Act: -3.069
ĠcheckĠn=5.ĠTheĠrootsĠofĠunityĠforĠn=5ĠareĠ1ĠandĠtheĠprimitiveĠ5

Layer 34

GATE_PROJ

Top 16 Positive Activations
Rollout 38, Act: 4.108
2Ġ-625rĠ+168=0.ĊĊUsingĠquadraticĠformula:ĊĊr=(62
Rollout 53, Act: 3.900
ĠinĠtheĠratioĠ2:1.ĠSoĠCP:PEĠ=Ġ2:1.ĠSinceĠPEĠis
Rollout 96, Act: 3.821
1=5αĠ-12βĊĊSolvingĠtheseĠnumerically:ĊĊFromĠfirstĠequation,Ġ-2
Rollout 96, Act: 3.807
697*895ĊĊAddingĠthemĠtoĠeliminateĠβ:ĊĊ(2316*2
Rollout 61, Act: 3.794
.ĠCheckĠ17:Ġ133ĠdividedĠbyĠ7ĠisĠ19;Ġyep
Rollout 28, Act: 3.790
4a2y2Ġ+8abĠxy.ĊĊCompareĠtoĠourĠequation:ĊĊ4b2x2+
Rollout 53, Act: 3.786
ĠalongĠtheĠmedianĠfromĠAĠtoĠD.ĠSoĠAP:PDĠ=Ġ2:1.ĠPDĠisĠgiven
Rollout 9, Act: 3.770
ĠisĠ100Ġ-500/h=80.ĠThenĠ500/h=
Rollout 81, Act: 3.759
Ġ-Ġ561Ġ=Ġ0ĊĊLet'sĠattemptĠtoĠsolveĠthisĠquadraticĠequationĠforĠh.ĠQuadr
Rollout 69, Act: 3.753
Ġ4Ġsin^2Ġtheta)ĊĊSo,Ġlet'sĠsubstituteĠthisĠintoĠterm_i:ĊĊsin2(theta_i
Rollout 9, Act: 3.747
Ġ(10,0):ĊĊUsingĠpoint-slopeĠform:ĠyĠ-0Ġ=Ġ(-5/h)(x
Rollout 81, Act: 3.723
Ġ-140hĠ-1122=0.ĊĊButĠbefore,ĠitĠwasĠ22
Rollout 86, Act: 3.719
2.ĠThat'sĠstraightforward:Ġ[x^5Ġ/5]ĠfromĠ0ĠtoĠpi/2Ġ=
Rollout 23, Act: 3.712
ĠPossiblyĠconsideringĠBatemanĠequationĠorĠparent-daughter,ĠbutĠthatĠdoesn'tĠapplyĠhereĠasĠYĠandĠQĠare
Rollout 62, Act: 3.711
55.ĠNot.Ġ2561ĠdividedĠbyĠ17:Ġ17*1
Rollout 7, Act: 3.707
PĠ=Ġ56NĠ+15P=66NĊĊSoĠ15P=
Top 16 Negative Activations
Rollout 71, Act: -2.179
ĠButĠ3c2ĠisĠevenĠifĠcĠisĠevenĠandĠoddĠotherwise.ĠSimilarĠforĠk2.ĠSince
Rollout 33, Act: -2.191
ĠthereĠshouldĠbeĠatĠleastĠoneĠnumberĠgreaterĠthanĠ2,ĠwhichĠmeansĠtheyĠcan'tĠbothĠbeĠ1
Rollout 75, Act: -2.194
Ġuniversally.ĊĊWait.ĊĊActually,ĠifĠweĠhaveĠthreeĠevenĠnumbers.ĊĊWait,ĠtimeĠtoĠpause.ĠLetĠme
Rollout 19, Act: -2.201
Ġby10.ĠRemainder5.ĊĊn=6:Ġsum=441ĠdividedĠby11
Rollout 33, Act: -2.209
Ġcan'tĠbeĠ1.ĠThenĠforĠk=2,ĠinĠtheĠfirstĠtwoĠelements,ĠthereĠshouldĠbe
Rollout 2, Act: -2.210
^1Ġ-1),ĠtheĠsecondĠisĠk=2Ġ(10^2Ġ-1),Ġ...,Ġup
Rollout 75, Act: -2.214
ĠsumĠofĠtheĠsquaresĠofĠanyĠthreeĠpairwiseĠdifferentĠpositiveĠoddĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquares
Rollout 75, Act: -2.217
ĠSoĠweĠneedĠthreeĠofĠtheĠsixĠsquaresĠtoĠbeĠoddĠ(1ĠmodĠ4)ĠandĠthreeĠtoĠbe
Rollout 75, Act: -2.230
ĠsumĠofĠtheĠsquaresĠofĠanyĠthreeĠpairwiseĠdifferentĠpositiveĠoddĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquares
Rollout 83, Act: -2.267
-ĠResidueĠ0:Ġ{3,Ġ6,Ġ9}Ċ-ĠResidueĠ1:
Rollout 75, Act: -2.305
ĠsumĠofĠtheĠsquaresĠofĠanyĠthreeĠpairwiseĠdifferentĠpositiveĠoddĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquares
Rollout 75, Act: -2.306
Ġ(1ĠmodĠ4)ĠandĠthreeĠtoĠbeĠevenĠ(0ĠmodĠ4).ĠTherefore,ĠinĠorder
Rollout 93, Act: -2.367
Ġfrom1Ġto1007.ĠForĠkĠeven,ĠT_{1007}(x_k
Rollout 58, Act: -2.406
ĠtoĠkĠofĠ(i/(i+1))^i.ĊĊSimilarly,ĠP(1)ĠisĠtheĠprobability
Rollout 33, Act: -2.492
k$ĠtermsĠofĠtheĠpermutationĠisĠgreaterĠthanĠ$k$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,
Rollout 75, Act: -2.613
Ġeligible?ĊĊWait,ĠconsiderĠevenĠnumbersĠsquared:ĊĊEachĠevenĠsquareĠisĠaĠmultipleĠof4.ĊĊTherefore,Ġsum

UP_PROJ

Top 16 Positive Activations
Rollout 25, Act: 1.867
=5.97e24Ġkg,ĠthenĠmâīĪ1.027e25
Rollout 60, Act: 1.746
_observerĠisĠtheĠspeedĠofĠtheĠobserverĠ(car)ĠtowardsĠtheĠsource.ĊĊButĠactually,ĠtheĠformulaĠdependsĠon
Rollout 47, Act: 1.698
ĠÏģĠv_air2ĊĊThus,ĠtheĠpressureĠdifferenceĠp0Ġ-ĠpĠ=Ġ(1/2)
Rollout 96, Act: 1.589
169))/5ĊĊLet'sĠcomputeĠnumerator:ĊĊ90Ġ=Ġ(90*169
Rollout 30, Act: 1.583
ĠtheĠinitialĠconditions,ĠwhichĠisĠaĠgoodĠsign.ĊĊTherefore,ĠthisĠfractionĠisĠindeedĠtheĠanswer.ĊĊButĠis
Rollout 99, Act: 1.572
Ġf(x1),Ġf(x2),ĠcomputeĠtheĠdifferences:ĊĊh0Ġ=Ġx1Ġ-Ġx0ĊĊ
Rollout 8, Act: 1.514
ĠoriginalĠproblem,ĠseemsĠthatĠmyĠanswerĠisĠright.ĊĊTherefore,ĠprobabilityĠisĠ31/126
Rollout 64, Act: 1.498
ĠthenĠk=0,ĠsoĠB=(a2Ġ-a1)/2=0,ĠandĠA=
Rollout 77, Act: 1.471
.ĠSoĠtheĠanswerĠisĠ364.ĊĊTherefore,ĠtheĠfunctionĠfĠisĠactuallyĠtheĠleastĠcommonĠmultiple
Rollout 99, Act: 1.468
Ġ=Ġ(f(x3)Ġ-Ġf(x2))/h1Ġ=Ġ(1.1765
Rollout 45, Act: 1.453
(-1)Ġ=Ġ-9Ġ+b,Ġso:ĊĊf(-2)Ġ=Ġ2(-9Ġ+
Rollout 88, Act: 1.451
.ĠSoĠtheĠanswerĠisĠ100.ĠTherefore,ĠIĠthinkĠit'sĠ100.
Rollout 26, Act: 1.442
Ġ122.64ĠkJ.ĊĊTherefore,ĠtheĠreactionĠasĠwrittenĠ(usingĠHg(g
Rollout 30, Act: 1.434
4Ġ)ĊĊHence,ĠtheĠresultĠisĠcorrect.ĊĊTherefore,Ġ1Ġ+1/(2015
Rollout 44, Act: 1.433
*V1),ĠsoĠnRTĠln(V1/V2)Ġ=Ġp1*V1Ġln(V
Rollout 73, Act: 1.413
ĠisĠ0),ĠsoĠit'sĠtheĠsame.ĊĊSoĠsum_{k=1}^{n}k
Top 16 Negative Activations
Rollout 86, Act: -2.558
2.ĠThat'sĠstraightforward:Ġ[x^5Ġ/5]ĠfromĠ0ĠtoĠpi/2Ġ=
Rollout 1, Act: -2.561
ĠofĠthem.ĊĊAlternatively,ĠforĠtheĠsystemĠtoĠhaveĠaĠsolution,ĠthenĠforĠanyĠx,Ġx'Ġin
Rollout 18, Act: -2.576
3,8),Ġ(4,-1),ĠandĠ(5,4)ĠisĠtheĠsameĠasĠtheĠsubgroup
Rollout 15, Act: -2.599
ĠlampsĠwillĠeventuallyĠturnĠoffĠafterĠsomeĠtime.ĊĊFirst,ĠletĠmeĠrephraseĠtheĠrulesĠtoĠmakeĠsureĠI
Rollout 45, Act: -2.600
4mĠforĠallĠnon-negativeĠintegersĠx.ĠFindĠtheĠgreatestĠpossibleĠvalueĠofĠm.ĊĊWait,ĠbutĠf
Rollout 24, Act: -2.607
ĠanĠassociatedĠsystemĠofĠinequalitiesĠthatĠdetermineĠRĠandĠC,ĠwhichĠareĠstructurallyĠsimilarĠacrossĠdifferentĠminimalĠpairs.
Rollout 1, Act: -2.610
ĠwithĠequalĠnormsĠandĠtheirĠdifferencesĠhaveĠequalĠnorms,ĠthisĠoftenĠrelatesĠtoĠthemĠbeingĠverticesĠofĠaĠsimplexĠor
Rollout 36, Act: -2.613
ĠequationsĠwithĠvariablesĠx,Ġy,Ġz,ĠandĠs.ĠWeĠneedĠtoĠsolveĠtheseĠequationsĠtoĠfindĠx
Rollout 85, Act: -2.622
,y)=ĠaĠxĠ+ĠbĠy,ĠwithĠaĠandĠbĠbeingĠ1ĠandĠ2ĠinĠsomeĠorder
Rollout 18, Act: -2.645
=H_{xy},ĠweĠneedĠtoĠdetermineĠxĠandĠyĠsuchĠthat:ĊĊ1.ĠTheĠgeneratorsĠofĠH
Rollout 51, Act: -2.645
ĠÏĢ:ĊĊâĪ«0^ÏĢĠcosnxĠdxĠ=Ġ[sin(nx)/n]0^ÏĢĠ=Ġ(
Rollout 37, Act: -2.668
ĠinherentĠsymmetryĠthatĠgroupsĠtheseĠtermsĠintoĠmultipletsĠwhichĠaĠcommonalityĠleadingĠtoĠdivisibility.ĠForĠexample,
Rollout 1, Act: -2.729
ĠorthogonalĠtoĠtheĠsetĠSĠ-ĠS,ĠexceptĠforĠaĠconstant.ĊĊButĠinĠthatĠcase,ĠifĠweĠtake
Rollout 51, Act: -2.737
x)ĠtimesĠtheĠabsoluteĠvalueĠofĠsin(xĠ-Ġθ)Ġdx,ĠwhereĠθĠisĠbetweenĠ
Rollout 8, Act: -2.785
Ġavailable.ĊĊTherefore,ĠthisĠisĠsimilarĠtoĠpermutationĠwithĠtwoĠcolors.ĠTheĠnumberĠofĠsuchĠsequencesĠisĠC(
Rollout 14, Act: -2.796
Ġ*1?ĠNotĠsureĠeither.ĊĊAlternatively,ĠuseĠhomogeneousĠfunctions.ĠSupposeĠfĠisĠaĠscalingĠfunctionĠsuchĠthat

DOWN_PROJ

Top 16 Positive Activations
Rollout 87, Act: 2.914
Ġ/Ġsqrt(η2Ġ+ĠÎ2))ĠevaluatedĠfromĠ0ĠtoĠ1.ĊĊ=Ġsinh^{-
Rollout 86, Act: 2.899
Thus,ĠintegralĠ=Ġ-x^3ĠcosĠxĠ|ĠfromĠ0ĠtoĠpi/2Ġ+Ġ3
Rollout 87, Act: 2.874
1(zĠ/Ġsqrt(x2Ġ+Ġy2))ĠfromĠ0ĠtoĠaĊĊWhichĠisĠequalsĠsinh^{-
Rollout 86, Act: 2.805
Ġpi/2ĠisĠ(1/2)x2ĠevaluatedĠfromĠ0ĠtoĠpi/2,ĠwhichĠis
Rollout 14, Act: 2.772
(Ġ\intĠf(x)^2ĠdxĠ-ĠCĠ\right)Ġ-Ġ\muĠ\left(Ġ\int
Rollout 14, Act: 2.761
Ġintegral:ĊĊâĪ«ĠxĠdxĠ=Ġx2Ġ/2ĠfromĠ0ĠtoĠ1/μ:Ġ(1
Rollout 86, Act: 2.721
x2)(1/2)Ġsin2xĠ|Ġfrom0ĠtoĠpi/2Ġ-ĠâĪ«Ġ(
Rollout 86, Act: 2.716
Ġ(1/2)x^2Ġsin2xĠ|Ġ-ĠâĪ«xĠsin2xĠdxĠ=0
Rollout 51, Act: 2.703
x/9Ġ+Ġcos7x/7]ĠevaluatedĠfromĠ0ĠtoĠÏĢ.ĊĊComputeĠatĠÏĢ:ĊĊ
Rollout 86, Act: 2.643
:Ġ(1/2)x2Ġsin2xĠ|Ġfrom0ĠtoĠpi/2Ġ=0.ĊĊThen
Rollout 9, Act: 2.517
10Ġ-100/h)*xĠ]ĠfromĠ0Ġto10.ĊĊEvaluatingĠat
Rollout 51, Act: 2.475
ĠtoĠpiĠisĠ[-cos9x/9]ĠfromĠ0ĠtoĠpiĠwhichĠisĠ(-cos9pi
Rollout 86, Act: 2.364
Ġx.ĊĊThus,ĠintegralĠ=Ġx3ĠsinĠxĠ|ĠfromĠ0ĠtoĠpi/2Ġ-Ġ3
Rollout 86, Act: 2.361
.ĊĊThus,Ġ3Ġ[Ġx2ĠsinĠxĠ|Ġfrom0ĠtoĠpi/2Ġ-ĠâĪ«Ġ
Rollout 86, Act: 2.359
xĠdxĠfrom0ĠtoĠpi/2=sinxĠ|0ĠtoĠpi/2=1Ġ-0=
Rollout 86, Act: 2.330
Integral=Ġ(x/2)Ġsin2xĠ|0ĠtoĠpi/2Ġ-Ġ(1/2
Top 16 Negative Activations
Rollout 78, Act: -2.612
9/800)Ġ=Ġ(22/3)Ġ*Ġ(1/800)
Rollout 14, Act: -2.615
?ĠSinceĠHölderĠcanĠrelateĠmultipleĠintegralsĠwithĠexponentsĠthatĠareĠconjugate.ĠLetĠmeĠrecallĠHö
Rollout 55, Act: -2.659
ĠrotationĠblockĠwouldĠbeĠrotatedĠbyĠ2*(ÏĢ/6)Ġ=ĠÏĢ/3.ĠA^3
Rollout 14, Act: -2.670
/2}Ġdx.ĊĊApplyĠHölderĠwithĠexponentsĠp=2,Ġq=4,ĠandĠr
Rollout 14, Act: -2.680
'sĠinequalityĠforĠthreeĠfunctionsĠwouldĠstateĠthatĠforĠexponentsĠ$p,Ġq,Ġr$ĠsuchĠthatĠ$
Rollout 96, Act: -2.706
90Ġ=Ġ(90*169)/169Ġ=Ġ15210
Rollout 14, Act: -2.734
.ĊĊLetâĢĻsĠrememberĠthatĠHölderâĢĻsĠinequalityĠwithĠexponentsĠ2ĠandĠ2ĠgivesĠCauchy
Rollout 53, Act: -2.737
3Ġ)Ġ/2Ġ+Ġ(-27/(2Ġsqrt73Ġ))Ġ=ĠsameĠasĠ73/(
Rollout 14, Act: -2.752
3/2?ĠWait,ĠHölder'sĠwithĠexponentsĠp,ĠqĠsuchĠthatĠ1/pĠ+
Rollout 14, Act: -2.776
}Ġdx.ĊĊThenĠapplyĠHölder'sĠinequalityĠwithĠexponentsĠp,Ġq,ĠrĠsuchĠthatĠ1
Rollout 14, Act: -2.823
LetâĢĻsĠrememberĠthatĠHölderâĢĻsĠinequalityĠwithĠexponentsĠ2ĠandĠ2ĠgivesĠCauchy-S
Rollout 14, Act: -2.835
Ġdx.ĊĊThenĠapplyĠHölder'sĠinequalityĠwithĠexponentsĠp,Ġq,ĠrĠsuchĠthatĠ1/p
Rollout 36, Act: -2.837
*(s2Ġ+Ġ8064)/(2s)Ġ=Ġ-Ġ(s2Ġ+Ġ8
Rollout 53, Act: -2.853
Ġsqrt73Ġ/2Ġ-27/(2Ġsqrt73Ġ)Ġ=Ġ(73Ġ-27
Rollout 14, Act: -2.856
lder'sĠinequalityĠforĠthreeĠfunctionsĠwouldĠstateĠthatĠforĠexponentsĠ$p,Ġq,Ġr$ĠsuchĠthat
Rollout 14, Act: -2.972
ĠthreeĠfunctionsĠf,Ġg,Ġh,ĠandĠexponentsĠp,Ġq,ĠrĠwithĠ1/pĠ+

Layer 35

GATE_PROJ

Top 16 Positive Activations
Rollout 26, Act: 4.381
0ĠhasĠfourĠsigĠfigs,Ġ312.8ĠhasĠfour,ĠtheirĠdifferenceĠisĠ1
Rollout 68, Act: 4.226
âīĪ[Ġ-504Ġ+Ġ503.98413Ġ]Ġ/2âīĪ
Rollout 68, Act: 4.167
Ġ/2âīĪ(-504Ġ+503.984)/2âīĪ(-0.
Rollout 17, Act: 4.137
T,ĠwhichĠwouldĠbeĠapproximatelyĠ25.4ĠmT,ĠsoĠ0.025
Rollout 81, Act: 4.047
2601*(533-49)=0.ĊĊĠCompute:ĊĊ533-4
Rollout 53, Act: 4.034
ĠbeĠanswer.ĊĊHence,ĠtheĠareaĠisĠ\boxed{\dfrac{27}{2}}.ĊĊ
Rollout 0, Act: 4.028
ĠanswerĠisĠ128,ĠcorrespondingĠtoĠ\boxed{128}?ĠĠOrĠwait,Ġwait
Rollout 68, Act: 4.021
[Ġ-504Ġ+ĠapproxĠ504Ġ]Ġ/2âīĪĠsomethingĠlikeĠ(-504
Rollout 68, Act: 4.021
ĠwouldĠbeĠ[Ġ-504Ġ-503.98413]/2âīĪ[-
Rollout 4, Act: 4.009
2Ġ+Ġ(2Ġ*Ġ0.004)^2Ġ)ĊĊCalculatingĠ2*0.
Rollout 81, Act: 3.994
2601*(533Ġ-49)=0.ĊĊComputeĠcoefficients:ĊĊComputeĠfirstĠcoefficient:
Rollout 40, Act: 3.981
ĠstepsĠcheckĠout.ĠTherefore,ĠtheĠanswerĠisĠ\boxed{19}.ĊĊ**FinalĠAnswer**Ċ\
Rollout 45, Act: 3.979
1Ġ+4=Ġ[(-1Ġ+4Ġ-4)]*2Ġ+Ġ(-6)Ġ+8Ġ+
Rollout 19, Act: 3.976
Ġis78.ĠTherefore,ĠboxĠitĠasĠ\boxed{78}.ĠButĠletĠmeĠtryĠk=
Rollout 32, Act: 3.968
18.ĠTherefore,ĠtheĠanswerĠisĠboxedĠ\boxed{4018}.ĊĊButĠbeforeĠfinal
Rollout 4, Act: 3.951
90=4975*(1200-10)=4975*
Top 16 Negative Activations
Rollout 87, Act: -2.712
ĠdξĠdηĠdζĊĊToĠcomputeĠK,ĠweĠcanĠlookĠforĠpossibleĠsolutions.ĠIĠmightĠhave
Rollout 25, Act: -2.728
ĠmĠ)/v2ĊĊSolvingĠforĠm,ĊĊm=Ġ(3.541e6
Rollout 1, Act: -2.746
Ġ||y||2Ġ-d2/2.ĊĊTherefore,ĠeachĠinnerĠproductĠofĠxiĠwithĠyĠisĠthis
Rollout 13, Act: -2.758
ĠtoĠmultiplyĠdirectly.ĠStill,ĠthisĠmightĠtakeĠtime.ĊĊAlternatively,ĠusingĠcurrencyĠcalculationĠsteps:ĊĊ1.5
Rollout 14, Act: -2.772
λ),ĠandĠzeroĠelsewhere.ĊĊComputeĠtheĠintegrals:ĊĊJĠ=ĠâĪ«0^{x_max}Ġ[(
Rollout 31, Act: -2.782
}Ġ-d_{m+3}=0.ĊĊTherefore,ĠthisĠgivesĠusĠlinearĠequationsĠforĠtheĠcoefficientsĠc
Rollout 36, Act: -2.788
xĠ+Ġ12600/sĊĊTherefore:ĊĊs2Ġ-Ġs(yĠ+Ġz)Ġ=
Rollout 51, Act: -2.798
2ÏĢĠcos8xĠ|sinx|ĠdxĊĊNow,ĠconsideringĠtheĠintegralĠoverĠ0ĠtoĠ
Rollout 87, Act: -2.814
atively.ĠLetĠmeĠtryĠtoĠcomputeĠKĠasĠfollows:ĊĊKĠ=ĠâĪ«0^1ĠâĪ«0
Rollout 87, Act: -2.839
1/rĠoverĠaĠcubicĠvolume.ĠThatĠseemsĠchallenging.ĊĊAlternatively,ĠisĠthereĠaĠknownĠintegralĠforĠpotentialĠat
Rollout 30, Act: -2.863
n+1}Ġ-1Ġ)Ġ-1.ĊĊTherefore,ĠtheĠtermĠweĠhaveĠinĠtheĠsum:ĊĊ[
Rollout 87, Act: -2.888
2)Ġdx.ĠLetĠmeĠcomputeĠthisĠintegralĠfirst.ĊĊLetĠmeĠdefineĠf(y,z)Ġ=ĠâĪ«
Rollout 4, Act: -2.897
ĊĊHmm,Ġlet'sĠcomputeĠtheseĠstepĠbyĠstep.ĊĊFirst,ĠcomputeĠtheĠcoefficientsĠforĠÎĶC_eq.ĊĊ
Rollout 14, Act: -2.900
1Ġ-ĠμĠx)/(2Ġλ)]Ġdx.ĊĊSoĠfirst,Ġlet'sĠcomputeĠJ:ĊĊLetâĢĻsĠdo
Rollout 73, Act: -3.004
,ĠperhapsĠnot.ĠMaybeĠIĠneedĠaĠbetterĠapproach.ĊĊAlternatively,Ġlet'sĠcomputeĠSĠ=Ġsum_{j
Rollout 89, Act: -3.100
/5)TĠ=Ġ100.ĊĊSoĠTĠ=Ġ100Ġ*Ġ(5/

UP_PROJ

Top 16 Positive Activations
Rollout 7, Act: 2.641
Ġ?ĊĊWait,Ġwait:ĊĊWaitĠ18NĠisĠequalĠtoĠ54N/3.ĠSo
Rollout 26, Act: 2.477
ĠkJ/KĠ(4ĠsigĠfigs)ĊĊ9-ĠÎĶT=14.8ĠKĠ(3
Rollout 28, Act: 2.425
,ĠIĠrememberĠthatĠaĠconicĠisĠdefinedĠbyĠfiveĠpoints,ĠbutĠhereĠweĠhaveĠfourĠpoints:ĠO
Rollout 30, Act: 2.410
*2014=4,058,210.ĠTherefore,ĠtheĠtotalĠsum
Rollout 75, Act: 2.375
ĠtheĠproblemĠsaysĠsixĠsquares.ĠSoĠaboveĠapproachĠgivesĠsevenĠsquares,ĠwhichĠisĠmoreĠthanĠallowed.ĠSoĠneed
Rollout 32, Act: 2.351
ĠÏī3Ġ=Ġ1,ĠsoĠÏī4Ġ=ĠÏī.ĠTherefore,ĠÏī4Ġ-
Rollout 70, Act: 2.283
ĠbutĠweĠhadĠN=7kĠ+d0,ĠandĠd0=2.ĠThus,ĠN=
Rollout 17, Act: 2.281
Wait,ĠweĠcanceledĠoutĠtheĠ10^-33ĠinĠnumeratorĠandĠdenominator,ĠsoĠtheĠexponentĠis
Rollout 30, Act: 2.238
5*2014=4,058,210.ĠTherefore,ĠtheĠtotal
Rollout 1, Act: 2.233
Ġ2Ġ=Ġ3d2/4Ġ-ĠwhichĠisĠproblematic.ĊĊHowever,ĠatĠtheĠsameĠtime,
Rollout 64, Act: 2.224
}-1Ġ)Ġ)ĊĊButĠ2^{2012}=8*2^{2009
Rollout 6, Act: 2.190
ĠareĠinĠĠ$\textbf{P}_{angel}$ĠĠbecauseĠtheĠ\textit{angelĠstring}
Rollout 64, Act: 2.178
ĠsoĠweĠcanĠsubstitute:Ġ2^{2012}Ġ=8*2^{200
Rollout 22, Act: 2.172
.ĊĊRĠisĠtheĠgasĠconstant,ĠwhichĠisĠ8.314462618
Rollout 75, Act: 2.162
ĠsquaresĠandĠ3Ġones.ĠWhichĠisĠsevenĠsquares.ĠNo.ĊĊWait,ĠtheĠproblemĠsaysĠsixĠsquares.
Rollout 68, Act: 2.153
*508)=sqrt(254000).ĠHmmm.ĊĊButĠregardless,Ġthe
Top 16 Negative Activations
Rollout 0, Act: -2.342
a}{b}\)ĠwhereĠ\(0Ġ<ĠaĠ<Ġb\)ĠandĠ\(a\)ĠandĠ\(b\)
Rollout 28, Act: -2.342
0):ĊĊabĠ=Ġ-2kĠa2Ġ/Ġ(2hĠ-a)Ġ-2hĠb2
Rollout 9, Act: -2.364
.ĠHowever,ĠanĠisoscelesĠtriangleĠwithĠbaseĠEM:ĠifĠit'sĠvertical,ĠthenĠtheĠbase
Rollout 65, Act: -2.366
.ĠThenĠtheĠoriginalĠlengthĠaĠisĠcutĠintoĠtwoĠparts,ĠsayĠxĠandĠaĠ-Ġx,ĠandĠthen
Rollout 40, Act: -2.376
ĊĊSecondĠterm:Ġ-10ĠÏī^n/(1Ġ-ĠÏī)Ġ+Ġ10/(
Rollout 41, Act: -2.385
ĠcountĠseparatelyĠforĠeachĠcase.ĠThatĠis,ĠcountĠallĠ1-digitĠnumbers,Ġ2-digitĠnumbers,
Rollout 56, Act: -2.396
ĠaĠright-angledĠtriangleĠatĠA0,ĠwithĠangleĠatĠA1ĠbeingĠ60°,Ġwhich
Rollout 41, Act: -2.418
ĠotherĠdigitĠisĠ'0'.ĊĊSoĠforĠeachĠ'a'Ġ(fromĠ1-9),ĠweĠcan
Rollout 5, Act: -2.430
125(xĠ+Ġ175)]/(xĠ+Ġ75)Ġ=Ġ12
Rollout 41, Act: -2.442
ĠcountĠtheĠnumberĠofĠthree-digitĠnumbersĠthatĠuseĠbothĠaĠandĠb,ĠarrangedĠinĠsomeĠorder.ĠEachĠposition
Rollout 70, Act: -2.487
ĠasĠmanyĠdigitsĠasĠpossible,ĠwithĠtheĠhighestĠpossibleĠdigits.ĊĊAlternatively,ĠperhapsĠit'sĠbetterĠtoĠfindĠN
Rollout 16, Act: -2.520
ĠweĠcanĠuseĠtheĠformulaĠthatĠforĠtheĠmaximumĠofĠindependentĠrandomĠvariables,ĊĊE[T]Ġ=Ġsum_{k
Rollout 33, Act: -2.570
=2,ĠtheĠfirstĠtwoĠpositionsĠmustĠnotĠbothĠbeĠfromĠ{1,2}.ĠWait,ĠbecauseĠif
Rollout 91, Act: -2.587
ĠC.ĊĊToĠfindĠangleĠatĠB:ĠcomputeĠtheĠangleĠbetweenĠBAĠandĠBC.ĊĊSimilarly,ĠtheĠangleĠat
Rollout 16, Act: -2.676
ĠcanĠbeĠtricky.ĠTheĠexpectationĠofĠtheĠmaximumĠofĠindependentĠgeometricallyĠdistributedĠvariablesĠisn'tĠstraightforward.ĠInĠthe
Rollout 5, Act: -2.947
Ġ+Ġ175)(125H)/(75Ġ+Ġx)Ġ=Ġ125

DOWN_PROJ

Top 16 Positive Activations
Rollout 91, Act: 3.465
Ġ(tĠ-1Ġ)*0Ġ-Ġ(-tĠ)*(-2Ġ)Ġ|Ġ=Ġ|0Ġ-2tĠ|=
Rollout 45, Act: 3.295
1)(-2)Ġ+Ġb*(-1)(-1)Ġ+Ġc*(-2)Ġ+4ĊĊ
Rollout 91, Act: 3.272
(tĠ-1Ġ)*0Ġ-Ġ(-tĠ)*(-2Ġ)Ġ|=Ġ|-2tĠ|Ġ=2t
Rollout 68, Act: 3.010
4=Ġ4-8Ġ+2kĠ+4=Ġ2k.ĠSoĠf(-2)=
Rollout 45, Act: 2.938
1)(-2)Ġ+8*(-1)(-1)Ġ+0*(-2)Ġ+4=
Rollout 81, Act: 2.918
3)(17)Ġ+Ġ(âĢĵh)(âĢĵh)=Ġ-51Ġ+Ġh2ĊĊ|
Rollout 55, Act: 2.863
ĠwouldĠbeĠ0*0Ġ+Ġ(-1)*(-1)Ġ+Ġ0*0Ġ=Ġ1
Rollout 90, Act: 2.844
Ġdigits,ĠMĠisĠoneĠdigit,ĠandĠLĠisĠfourĠdigits.ĠForĠexample,ĠN=100
Rollout 81, Act: 2.821
(-3Ġ-Ġ0)Ġ=Ġ-hĠ/Ġ(-3)Ġ=Ġh/3.ĊĊSimilarly,Ġthe
Rollout 96, Act: 2.812
69Ġ=27Ġ-Ġ(-12)*(-526)/169ĊĊComputeĠx
Rollout 81, Act: 2.808
17)+Ġ(-51/11)(-51/11)=Ġ-51Ġ+
Rollout 91, Act: 2.796
tĠ+1)(2)Ġ+Ġ(t)(0)Ġ=Ġ2(tĠ+1)ĊĊ|BA
Rollout 90, Act: 2.786
9,Ġso:ĊĊNĠâī¡Ġ(-MĠ*Ġ(-112))ĠmodĠ1009
Rollout 17, Act: 2.735
6âīĪĠCalculate:ĊĊ400*3.1416=1256.6
Rollout 91, Act: 2.723
Ġ|(tĠ-1)(0)Ġ-Ġt(-2)|Ġ=Ġ|0Ġ+Ġ2t|
Rollout 91, Act: 2.699
{BC},$ĠthenĠfindĠtheĠvalueĠofĠ$|\cotĠBĠ-Ġ\cotĠC|.$Ċ<|im_start|>
Top 16 Negative Activations
Rollout 3, Act: -2.204
25,921;162^2=26,244.ĠSo
Rollout 47, Act: -2.205
2)ĊĊWeĠalreadyĠfoundĠv_airĠ=Ġ(4/3)Ġvt.ĠLet'sĠsubstitute:ĊĊp0
Rollout 89, Act: -2.207
ĠtheyĠflippedĠtheĠratio.ĠMaybeĠgirlsĠareĠ3/5.ĠLet'sĠimagine.ĠIfĠgirlsĠare3
Rollout 89, Act: -2.215
3ĠofĠtheĠnumberĠofĠboys?ĠBecauseĠ3/5Ġ=Ġboys,ĠsoĠperĠunitĠisĠ1
Rollout 84, Act: -2.229
Ġdrunk.ĠHence,ĠthatĠtheĠwastedĠisĠ2/9ĠofĠtheĠoriginalĠwastedĠamountĠinĠvolume.ĊĊGiven
Rollout 34, Act: -2.230
Ġx(t)Ġ-Ġ2ĠâĪ«_{0}^{t}Ġe^{-2(tĠ-Ġs)}
Rollout 73, Act: -2.231
ĠSĠhasĠ11Ġelements,ĠindicesĠ0ĠtoĠ10,ĠsoĠinĠourĠsum,Ġj
Rollout 32, Act: -2.251
atrix}Ċ\]ĊĊNow,ĠcomputeĠM2:ĊĊ\[Ċ\mathbf{M}^2Ġ=
Rollout 89, Act: -2.276
100,ĠthenĠgirlsĠwouldĠbeĠ(2/3)*100Ġ=Ġ66.
Rollout 95, Act: -2.298
ĠtheirĠareasĠareĠ(k/2450)^2Ġ*ĠAreaĠABC.ĠThen,ĠtheĠareaĠof
Rollout 8, Act: -2.301
5âĪĴr}ĠorĠG^{5âĪĴr}ĠR^r.ĠSoĠtwoĠoptions.ĊĊInĠterms
Rollout 7, Act: -2.303
3P=2NĠ=>ĠP=Ġ(2/3)N.ĊĊYesĠthat'sĠcorrect.ĠSo
Rollout 90, Act: -2.343
018.ĊĊ2018*100=201,800.
Rollout 34, Act: -2.371
Ġconstants:Ġ4Ġ-1Ġ-3Ġ=Ġ0ĊĊSo,ĊĊ=Ġe^{-2t}Ġ[
Rollout 73, Act: -2.375
1ĠtoĠn,ĠsoĠsubtractĠtheĠk=0ĠtermĠ(whichĠisĠ0),ĠsoĠit'sĠthe
Rollout 24, Act: -2.432
ĠallowĠaĠsmallerĠset.ĠHence,ĠtheyĠmustĠbeĠincomparable.ĊĊTherefore,Ġthere'sĠaĠpossibilityĠtoĠbuild

Layer 36

GATE_PROJ

Top 16 Positive Activations
Rollout 25, Act: 3.454
ĠisĠknown.ĊĊGivenĠthat,ĠproceedĠstepĠbyĠstep.ĊĊFirst,ĠstepĠ1:ĠsolveĠforĠrĠfrom
Rollout 53, Act: 3.423
Ġ)ĊĊLetĠmeĠwriteĠtheseĠwithĠvariablesĠforĠeasierĠcalculation.ĊĊComputeĠtheĠareaĠas:ĊĊFirst,ĠcomputeĠsumĠof
Rollout 96, Act: 3.392
697Ġ...(2)ĊĊSolveĠthisĠsystem.ĊĊItĠmightĠbeĠtediousĠbutĠletĠmeĠwriteĠthemĠdirectly
Rollout 14, Act: 3.366
Therefore,ĠdeterminingĠλĠandĠμĠtoĠsatisfyĠtheĠconstraints.ĊĊSoĠsupposeĠxĠrangesĠupĠtoĠx_max=1
Rollout 33, Act: 3.319
.ĊĊSoĠIĠneedĠtoĠcomputeĠeachĠofĠtheseĠterms.ĠLet'sĠstartĠanalyzingĠeachĠ|A_k|.ĊĊ
Rollout 56, Act: 3.286
.ĊĊTherefore,ĠthisĠgivesĠaĠrecursionĠforĠeachĠextension.ĊĊGivenĠthat,ĠperhapsĠtheĠseriesĠofĠareasĠcanĠbe
Rollout 96, Act: 3.254
Ġ1740Ġ/Ġ169.ĊĊWait,ĠletĠmeĠcomputeĠthat.ĠDivideĠnumeratorĠand
Rollout 52, Act: 3.244
ĠsmallerĠintegers.ĊĊSoĠlet'sĠproceedĠstepĠbyĠstep.ĠComputeĠforĠeachĠk,ĠandĠcheckĠifĠnĠis
Rollout 53, Act: 3.237
.ĠWeĠcanĠuseĠtheseĠequationsĠtoĠsolveĠforĠthem.ĊĊFirstly,ĠsubtractĠequationĠ3ĠfromĠequationĠ
Rollout 73, Act: 3.236
2^j.ĊĊLetĠmeĠevaluateĠeachĠtermĠseparately.ĊĊFirst,Ġsum_{j=1}^{1
Rollout 96, Act: 3.234
ĠseemsĠratherĠmessy.ĠLetĠmeĠcheckĠmyĠcomputationsĠagain.ĊĊEquation1:5xĠ-12y
Rollout 96, Act: 3.230
ĠneedĠtoĠsolveĠtheseĠtwoĠequationsĠforĠtĠandĠs.ĊĊMultiplyĠequation(1)Ġby12:1
Rollout 20, Act: 3.227
*43*47.ĊĊCalculatingĠthat:ĊĊMultiplyĠstepĠbyĠstep:ĊĊ2*3=6
Rollout 99, Act: 3.226
So,Ġlet'sĠcalculateĠtheseĠparametersĠstepĠbyĠstep.ĊĊGivenĠourĠdataĠpoints:ĊĊx0Ġ=Ġ0
Rollout 28, Act: 3.222
Ġ(3)^2*(3)^2=0Ġ?ĊĊWaitĠno,ĠwithĠa=3,Ġb=
Rollout 50, Act: 3.205
ĠtheĠcoefficientsĠofĠ\(t\)ĠandĠtheĠconstantĠterms.ĊĊFirst,Ġlet'sĠcomputeĠ\((\log_{
Top 16 Negative Activations
Rollout 6, Act: -0.904
}$Ġ.ĠShowĠthatĠthereĠexistsĠaĠsparseĠsetĠĠ$S_L$ĠĠandĠaĠdeterministicĠpolynomialĠtimeĠTM
Rollout 48, Act: -0.909
19EĠ(sinceĠ57,344ĠisĠ0xE000,Ġ
Rollout 39, Act: -0.914
ĠintegerĠ\[NĠ=Ġ9Ġ+Ġ99Ġ+Ġ999Ġ+Ġ999
Rollout 45, Act: -0.920
3Ġ+ĠmĠx^2Ġ+2mĠxĠyĠ+mĠy^2Ġ+nĠxĠ+n
Rollout 34, Act: -0.920
[Ċy(t)Ġ=Ġ-4Ġ+Ġe^{-t}Ġ+ĠCe^{t}Ċ\]ĊĊUsingĠthe
Rollout 57, Act: -0.924
Ġopen.ĠTherefore,ĠsinceĠUĠisĠopen,ĠitsĠcomplementĠCĠisĠclosed.ĠForĠtheĠclosedĠsetĠC,
Rollout 69, Act: -0.952
2Ġsin2ĠthetaĠ=Ġ1Ġ-ĠcosĠ2Ġtheta.ĠTherefore:ĊĊ1Ġ-Ġ4Ġsin2
Rollout 18, Act: -0.975
ĠZ2ĠâĨĴĠZ/xZĠdefinedĠbyĠÏĨ(k,Ġm)Ġ=ĠmĠ-ĠkĠyĠmodĠx.
Rollout 93, Act: -1.003
hevĠpolynomialsâĢĻĠextremaĠareĠatĠcos(kÏĢ/n)ĠforĠk=0,1,...,
Rollout 63, Act: -1.007
,Ġ1)Ġ=Ġf(1,Ġ3)Ġ=Ġ3Ġ\)Ċ-Ġ\(Ġf(
Rollout 74, Act: -1.018
ĠK_bĠmightĠbeĠtheĠcoefficientĠforĠtheĠfore-and-aftĠcomponent,ĠandĠK_sĠforĠtheĠathwart
Rollout 26, Act: -1.040
:ĊĊHg(l)Ġ+Ġ1âģĦ2ĠO2(g)ĠâĨĴĠHgO(s)ĊĊBut
Rollout 31, Act: -1.056
ĠrealĠnumbersĠ$x,Ġy$.ĠForĠanyĠpairĠofĠrealĠnumbersĠ$a_{0},Ġa_{1
Rollout 93, Act: -1.075
om{2014}{1007}Ġ+Ġ\binom{2014
Rollout 74, Act: -1.102
ĠbyĠexternalĠmagneticĠfields,ĠlikeĠthoseĠfromĠaĠship'sĠownĠmagnetismĠorĠnearbyĠferrousĠmaterials.ĠThese
Rollout 6, Act: -1.196
'sĠtheoremĠequivalenceĠbetweenĠ\(Ġ\textbf{P}/\text{poly}Ġ\)ĠandĠsparseĠor

UP_PROJ

Top 16 Positive Activations
Rollout 72, Act: 2.376
Ġboxed,ĠshouldĠbeĠ\boxed{2k}ĠforĠintegerĠkĠâī¥1,ĠbutĠsinceĠnĠmustĠbe
Rollout 95, Act: 2.367
TriangleĠ\(ABC\)ĠhasĠsideĠlengthsĠ\(ABĠ=Ġ9\),Ġ\(BCĠ=Ġ5\sqrt
Rollout 20, Act: 2.342
ĠisĠaĠmultipleĠofĠtheĠorderĠofĠpĠmoduloĠmĠforĠeveryĠprimeĠpĠâīłĠm.ĊĊButĠsinceĠthatĠmust
Rollout 89, Act: 2.313
ĠthatĠexactlyĠ\(Ġ\frac{2}{5}Ġ\)ĠofĠtheĠchildrenĠinĠaĠclassĠareĠgirlsĠand
Rollout 37, Act: 2.313
ĠproductĠofĠprimesĠupĠtoĠ2015ĠandĠtheirĠmultiplicities.ĠButĠperhapsĠforĠeachĠprimeĠp
Rollout 99, Act: 2.302
log_{10}(x)Ġ-ĠxĠ+Ġ3Ġ=Ġ0Ġ\)ĠwithĠinitialĠpointsĠ\(
Rollout 86, Act: 2.275
ĠeachĠfunction.ĠHowever,Ġactually,ĠtheĠselectionĠisĠwithĠrepetition,ĠsoĠallĠpermutationsĠofĠtheseĠcountsĠwouldĠbe
Rollout 58, Act: 2.264
<|im_start|>answerĊGivenĠanĠexamĠwithĠ\(ĠkĠ>Ġ1Ġ\)Ġparts,ĠwhereĠeachĠpartĠ\(Ġi
Rollout 19, Act: 2.261
ĠkĠ\)ĠifĠ2ĠandĠ\(ĠkĠ\)ĠareĠcoprime.ĠButĠ2ĠandĠ\(
Rollout 64, Act: 2.247
12=2009+3,ĠsoĠ2^{2012}Ġ-1
Rollout 60, Act: 2.217
ĠsourceĠemitsĠsoundĠofĠfrequencyĠ\(Ġf_0Ġ=Ġ492Ġ\,Ġ\text{Hz}
Rollout 99, Act: 2.204
ĠcalculateĠtheĠdifferencesĠandĠcoefficientsĠforĠtheĠquadraticĠpolynomial:Ċ\[Ċ\begin{align*}Ċh_0
Rollout 45, Act: 2.197
ĊGivenĠtheĠfunctionĠ\(Ġf:Ġ\mathbb{Z}Ġ\toĠ\mathbb{Z}
Rollout 65, Act: 2.191
Ġ\leqĠcĠ\)ĠwhereĠ\(ĠbĠ=Ġ1995Ġ\),ĠweĠneedĠtoĠdetermine
Rollout 97, Act: 2.156
.ĠTheĠrateĠconstantĠ\(k\)ĠisĠ\(5Ġ\timesĠ10^{-4}Ġ\,Ġ\
Rollout 64, Act: 2.149
\)ĠdefinedĠbyĠtheĠrecurrenceĠrelationĠ\(a_nĠ=Ġ3a_{n-1}Ġ-Ġ2
Top 16 Negative Activations
Rollout 36, Act: -2.205
Ġ0)Ċ-ĠD:Ġ(0,Ġ0,Ġs)Ċ-ĠG:ĠSinceĠAGĠisĠa
Rollout 36, Act: -2.213
(s,Ġ0,Ġ0)ĊĊC(0,Ġs,Ġ0)ĊĊD(0,
Rollout 9, Act: -2.215
5),Ġ(10,0),Ġ(10,10).ĠTheĠoverlappingĠareaĠbetweenĠthisĠtriangle
Rollout 81, Act: -2.230
ĠAĠatĠ(0,h)ĠandĠDĠatĠ(0,0),ĠthenĠBDĠisĠ3ĠandĠDC
Rollout 36, Act: -2.238
Ġ+Ġz^2Ġ=Ġ18000ĊĊâĪļ[x^2Ġ+Ġy^2
Rollout 56, Act: -2.246
0ĠtoĠA3Ġ[...]ĠandĠtheĠnewĠpastureĠisĠtriangleĠA1A2A3.ĠNext,Ġshe
Rollout 36, Act: -2.300
(0,Ġs,Ġ0)ĊĊD(0,Ġ0,Ġs)ĊĊG(s,Ġs
Rollout 9, Act: -2.304
Ġ(10,0)ĊMĠisĠ(10,10)ĊGĠisĠ(10Ġ-
Rollout 9, Act: -2.307
0,10),ĠandĠEĠatĠ(10,0).ĠThat'sĠconsistent.ĊĊNowĠtriangleĠG
Rollout 38, Act: -2.327
0),ĠsouthwestĠ(L,ĠW),ĠsoutheastĠ(0,ĠW).ĠThatĠmightĠbeĠcomplicatedĠbecauseĠxĠusually
Rollout 38, Act: -2.398
Ġ(L,W),ĠandĠsoutheastĠcornerĠisĠatĠ(0,W).ĠWait,ĠholdĠon:ĠSinceĠtheĠlonger
Rollout 36, Act: -2.407
Ġs,Ġ0)ĊĊD(0,Ġ0,Ġs)ĊĊG(s,Ġs,Ġs)ĊĊ
Rollout 9, Act: -2.445
ĠE=(10,0),ĠM=(10,10),ĠG=(10Ġ-Ġh
Rollout 36, Act: -2.482
Ġ-Ġ2syĠ+Ġs2Ġ=Ġ18000ĊĊEquationĠ(3):Ġx
Rollout 36, Act: -2.531
2Ġ+Ġz2Ġ=Ġ18000Ġ=>Ġ(x2Ġ+Ġy2Ġ+Ġz2
Rollout 36, Act: -2.537
syĠ+Ġs2Ġ=Ġ18000ĊĊEquationĠ(3):Ġx2Ġ+Ġy

DOWN_PROJ

Top 16 Positive Activations
Rollout 45, Act: 2.732
)Ġ=Ġ-1Ġ+2ĠâĩĴkĠ=1ĊĊTherefore,Ġk=1.ĊĊThenĠfromĠEquation4
Rollout 75, Act: 2.601
,102,42,22,12,32.ĠThisĠisĠeightĠsquares
Rollout 9, Act: 2.599
Ġ=Ġ100Ġ=>ĠyĠ=Ġ5.ĊĊTherefore,ĠtheĠy-coordinateĠofĠpointĠGĠmustĠbe
Rollout 62, Act: 2.565
ĠanotherĠnumberĠisĠsemiprimeĠ(twoĠprimes)ĠorĠtriprimeĠ(three),ĠthenĠtheĠoverallĠfactor
Rollout 93, Act: 2.516
03.5.ĠSoĠ(503.5Ġ+Ġ0.5)/100
Rollout 38, Act: 2.504
ĠfindĠtheĠratioĠofĠtheĠfield'sĠlengthĠtoĠitsĠwidthĠtoĠtheĠdistanceĠfromĠDĠtoĠtheĠsoutheastĠcorner.
Rollout 75, Act: 2.437
,102,42,22,12.ĠWhichĠisĠeightĠsquares,ĠneedĠto
Rollout 14, Act: 2.422
int_0^\inftyĠf(x)^2Ġdx$,ĊĊ$KĠ=Ġ\int_0^\
Rollout 75, Act: 2.422
2,42,22,12,32.ĠThisĠisĠeightĠsquares.ĠIfĠwe
Rollout 75, Act: 2.416
Ġ+102Ġ+42Ġ+22Ġ+12Ġ+32.ĠThat'sĠnineĠsquares
Rollout 95, Act: 2.400
/3ĠâīĪĠ8.333...ĊĊThenĠxĠ=Ġ25/3.ĠThen
Rollout 95, Act: 2.382
0/18Ġ=Ġ25/3ĠâīĪĠ8.333...ĊĊThenĠx
Rollout 23, Act: 2.370
70=10^(-30.413)=ĠaroundĠ10^-30
Rollout 53, Act: 2.360
.5Ġ/6Ġ=18.25ĊĊTherefore,Ġa=Ġsqrt(73/4
Rollout 8, Act: 2.357
ĠprobabilityĠthatĠKathyĠwillĠbeĠhappyĠisĠ$\frac{m}{n}$Ġ,ĠwhereĠ$m$ĠandĠ$
Rollout 14, Act: 2.324
1})^{1/(2aĠ-1)}}ĊĊFromĠK_λ=1:Ġλ^{a
Top 16 Negative Activations
Rollout 28, Act: -1.984
Ġ-Y2)ĊĊThirdĠterm:4*(X2Ġ+2XYĠ+Y2)/2=2
Rollout 20, Act: -1.987
Ġ1681Ġmod43:43*39=1677,Ġso
Rollout 6, Act: -1.998
ĠallowsĠLĠtoĠbeĠreduced.ĠSoĠprobablyĠtheĠintendedĠanswerĠisĠtoĠcite/toĠuseĠthisĠtheorem.ĊĊHence
Rollout 65, Act: -2.001
SetĠupĠratiosĠx/aĠ=Ġb/bĠ=Ġc/cĠâĩĴx/aĠ=1âĩĴx=a,Ġwhich
Rollout 19, Act: -2.028
Ġand83.ĠHence,ĠonlyĠdivisorsĠâī¥6Ġare83Ġitself.ĠTherefore,Ġk=8
Rollout 1, Act: -2.031
Ġcircular.ĊĊAlternatively,ĠsettingĠtĠ=Ġ||x||Ġsquared,ĠyouĠgetĠequations:ĊĊ-⣨x,Ġy
Rollout 45, Act: -2.039
1)xĠ+4mĠforĠallĠxĠâī¥0Ġintegers.ĊĊRewrittenĠasĠx3Ġ+4x2
Rollout 8, Act: -2.047
Ġactually,ĠforĠk=2,ĠisĠtheĠnumberĠ1200ĠperĠsequence?ĠWait,Ġno
Rollout 46, Act: -2.084
,ĠwhenĠtheyĠsayĠ"inĠtermsĠofĠU,"ĠtheyĠacceptĠexpressingĠitĠasĠtheĠgradientĠofĠUĠadjustedĠby
Rollout 6, Act: -2.111
,ĠbutĠhereĠtheĠadviceĠisĠfixedĠforĠallĠinputsĠofĠlengthĠn,ĠandĠtheĠmachineĠrunsĠinĠpoly-time
Rollout 3, Act: -2.173
2)^2Ġ=Ġ170^2Ġ-Ġ4*170Ġ+Ġ4Ġ=
Rollout 71, Act: -2.195
Ġac)xĠ-Ġabc=Ġx3Ġ-0x2Ġ+Ġ(-2011)xĠ+Ġ(-abc
Rollout 1, Act: -2.203
||2.ĠTherefore:ĊĊd2Ġ=Ġ(d2)/2Ġ+Ġ(d2)/2Ġ-Ġ
Rollout 83, Act: -2.241
,ĠandĠanyĠrotatedĠcopyĠdiffers.ĊĊInĠstandardĠBurnsideĠapproach,ĠwhenĠconsideringĠlabelingsĠunderĠrotationĠwithĠdistinct
Rollout 22, Act: -2.414
)^2Ġ=Ġ1,7002Ġ-Ġ2*1,700*6
Rollout 12, Act: -2.613
ĠifĠthatĠhelps.ĠAlternatively,ĠsubstitutionĠsĠ=Ġt2Ġ-Ġν2?ĊĊAlternatively,ĠsubstitutionĠÏĨĠwhereĠt

Layer 37

GATE_PROJ

Top 16 Positive Activations
Rollout 50, Act: 3.950
01)Ġ=Ġ11.ĊĊWhichĠchecksĠout.ĠTherefore,ĠtheĠvalueĠofĠ\(x\)Ġis
Rollout 5, Act: 3.781
5).ĠIsĠthatĠright?ĊĊWait,ĠletĠmeĠcheckĠtheĠcalculationsĠagainĠstepĠbyĠstep.ĠProbably,ĠI
Rollout 51, Act: 3.765
-\theta)|\Ġdx.Ġ\]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠintegral
Rollout 41, Act: 3.738
Ġ+Ġ576ĊĊLet'sĠcomputeĠthatĠstepĠbyĠstep:ĊĊFirst,Ġ9Ġ+90
Rollout 16, Act: 3.724
.032Ġ=Ġ6.193548...ĊĊk=6:ĠC(
Rollout 17, Act: 3.712
I}|$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackle
Rollout 67, Act: 3.704
1020=1848.ĠCorrect.ĊĊGood.ĠSoĠc_nĠisĠmodeledĠbyĠthis
Rollout 67, Act: 3.670
Ġ-420=1020.ĠCorrect.ĠSoĠequationĠ3ĠchecksĠoutĠgivenĠequationsĠ
Rollout 13, Act: 3.601
ĠaĠdollarĠvalueĠtoĠtheĠnearestĠcent.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠJoanieĠtookĠa
Rollout 92, Act: 3.598
(-0.615Ġm).ĠLet'sĠcomputeĠthis.ĠFirst,ĠconvertĠcentimetersĠtoĠmeters.ĊĊ
Rollout 59, Act: 3.593
.ĠFindĠ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠtheĠproblemĠisĠabout
Rollout 99, Act: 3.585
ĠtheĠdecimalĠvalueĠofĠx_3?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠwork
Rollout 13, Act: 3.582
.12^4.ĠLetĠmeĠdoĠthisĠstepĠbyĠstep.Ċ1.12^1Ġ=
Rollout 52, Act: 3.561
50=2014.ĠYep,Ġcorrect.ĠSoĠvalid.ĠSoĠtheĠsequenceĠfromĠ-1
Rollout 40, Act: 3.549
19.1421356.ĊĊSoĠtheĠgreatestĠintegerĠlessĠthanĠorĠequalĠtoĠ
Rollout 20, Act: 3.542
licative,ĠtheĠgeneralĠcaseĠwouldĠfollow.ĠLetĠmeĠverifyĠthat.ĊĊSupposeĠthatĠforĠeveryĠprimeĠpowerĠp
Top 16 Negative Activations
Rollout 66, Act: -2.511
2Ġ=Ġc2,ĠsoĠc2Ġ-Ġa2Ġ=Ġb2,ĠthusĠ(cĠ-Ġa)(
Rollout 37, Act: -2.524
ĠcanĠmodelĠtheĠdeterminantĠasĠtheĠproductĠofĠtheĠdifferencesĠmultipliedĠbyĠaĠsymmetricĠfunction.ĠThenĠuseĠtheĠfactĠthat
Rollout 77, Act: -2.527
)Ġ=Ġ\frac{52}{52Ġ-Ġ14}Ġ\cdotĠf(14
Rollout 66, Act: -2.532
ĠthatĠtheĠsidesĠwereĠa=Ġm2Ġ-n2,Ġb=2mn,ĠbutĠwhenĠinĠreality
Rollout 66, Act: -2.538
ĠwithĠoppositeĠparity,Ġa=Ġm2Ġ-n2,Ġb=Ġ2mnĠisĠgivenĠasĠa
Rollout 38, Act: -2.614
0,0).ĠTheĠlengthĠisĠLĠ(westĠdirection,ĠsoĠnegativeĠx-axis)ĠandĠwidthĠWĠ(
Rollout 12, Act: -2.621
ν}^1Ġ[Ġ(t2Ġ-Ġν2)^{-1/2}Ġ(1Ġ-Ġt
Rollout 66, Act: -2.684
ĠitsĠsidesĠareĠofĠtheĠformĠ\(m^2Ġ-Ġn^2\),Ġ\(2mn\),
Rollout 12, Act: -2.693
Ġexpression:ĠE(k)Ġ-Ġ(1Ġ-k2)ĠK(k).ĠIfĠthisĠisĠapproaching1Ġ-
Rollout 77, Act: -2.708
ĠexpressĠthisĠinĠtermsĠofĠf(a,ĠbĠ-Ġa)?ĠBecauseĠifĠIĠcanĠdoĠthat,ĠmaybeĠI
Rollout 38, Act: -2.724
ĠoriginĠ(0,0).ĠLetĠx-axisĠbeĠwestĠ(soĠnegativeĠdirectionĠisĠwest),Ġy-axisĠbe
Rollout 77, Act: -2.762
ĠweĠcanĠreplaceĠitĠwithĠ(a,ĠbĠ-Ġa).ĠRepeatingĠuntilĠweĠgetĠtoĠ(a,
Rollout 12, Act: -2.831
ν}^1Ġ(Ġ(t2Ġ-Ġν2)^{-1/2}Ġ(1Ġ-Ġt
Rollout 66, Act: -2.985
ĠĠĠ\[ĊĠĠĠ\frac{ab}{cĠ-Ġa}Ġ=Ġ6pĊĠĠĠ\]Ċ
Rollout 66, Act: -3.179
ĠofĠtheĠformĠ\(m^2Ġ-Ġn^2\),Ġ\(2mn\),ĠandĠ\(m
Rollout 66, Act: -3.189
ĠprimitiveĠtriplesĠviaĠtheĠformulaĠ(m2Ġ-n2,Ġ2mn,Ġm2Ġ+n2

UP_PROJ

Top 16 Positive Activations
Rollout 5, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊOneĠbaseĠofĠaĠtrapez
Rollout 34, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊFindĠcontinuousĠfunctionsĠĠ$x(t
Rollout 47, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠtrainĠwithĠcross-sectionalĠareaĠ$
Rollout 43, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠC-NOTĠgate
Rollout 78, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊJonĠandĠSteveĠrideĠtheirĠbicyclesĠalong
Rollout 63, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$P$ĠĠdenote
Rollout 60, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequency
Rollout 94, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊDetermineĠwhichĠsetĠofĠstatesĠmentioned
Rollout 84, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊGaryĠpurchasedĠaĠlargeĠbeverage,Ġbut
Rollout 55, Act: 3.582
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĊ\[\mathbf{
Rollout 32, Act: 3.575
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠ$\omega$ĠbeĠaĠnon
Rollout 40, Act: 3.575
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠparticleĠisĠlocatedĠonĠtheĠcoordinate
Rollout 35, Act: 3.575
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠ$P(x)$ĠbeĠa
Rollout 58, Act: 3.575
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊPatchouliĠisĠtakingĠanĠexam
Rollout 50, Act: 3.575
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠpositiveĠrealĠnumberĠ$
Rollout 59, Act: 3.575
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊJackieĠandĠPhilĠhaveĠtwoĠfair
Top 16 Negative Activations
Rollout 72, Act: -2.504
Ġ-Ġ1Ġ=Ġ0Ġ\),ĠsoĠtheĠsumĠisĠzero.Ċ9.ĠSimilarly,ĠcombinationsĠwithĠtwo
Rollout 71, Act: -2.505
SumĠofĠsquaresĠisĠΣa2=Ġ(Σa)^2Ġ-Ġ2Σab=Ġ
Rollout 27, Act: -2.514
1/2)mĠv_z2,ĠthenĠ⣨E⣩Ġ=Ġ(1/2)mĠ⣨v_z
Rollout 11, Act: -2.516
ĠpĠisĠequalĠtoĠtheĠsumĠofĠtheĠvariablesĠplusĠtheĠsumĠofĠtheirĠproductsĠtwoĠatĠaĠtime.ĠThat
Rollout 89, Act: -2.559
Ġ\(Ġ\frac{3}{5}Ġ\)ĠofĠtheĠchildrenĠareĠboys,ĠwhichĠweĠknowĠisĠ
Rollout 95, Act: -2.576
ĠsimilarĠtoĠABC,ĠwithĠratioĠx,ĠthenĠtheirĠareasĠareĠx2Ġ*ĠAreaĠABC.ĠButĠinĠthe
Rollout 11, Act: -2.587
ĠequalĠtoĠtheĠsumĠofĠtheĠvariablesĠplusĠtheĠsumĠofĠtheirĠproductsĠtwoĠatĠaĠtime.ĠThatĠformĠmakes
Rollout 85, Act: -2.587
.ĠTherefore,ĠinĠbothĠcases,ĠthereĠareĠtwoĠnumbersĠofĠoneĠparityĠandĠoneĠofĠtheĠother.ĠSo
Rollout 9, Act: -2.589
ĠbeĠequal.ĠTherefore,ĠtheĠtriangleĠhasĠtwoĠequalĠsidesĠ(GEĠandĠGM)ĠwithĠbaseĠEM.ĊĊSo
Rollout 76, Act: -2.590
ĠsquareĠ(2,2).ĊĊTherefore,ĠtheĠintersectionĠofĠallĠthreeĠisĠtheĠsingleĠsquareĠ(2,2
Rollout 95, Act: -2.624
ĠsimilarĠtoĠtriangleĠABC.ĠTherefore,ĠtheĠratioĠofĠareasĠwouldĠbeĠproportionalĠtoĠtheĠsquareĠofĠscalingĠfactor.ĊĊ
Rollout 86, Act: -2.630
xĠdx=Ġ-pi/4.ĠTherefore,Ġ-âĪ«xĠsin2xĠdxĠ=Ġ-(-pi
Rollout 89, Act: -2.689
=ĠTĠdividedĠbyĠ5,Ġgirls=2Ġunits,Ġboys=3Ġunits.ĠTherefore,ĠperĠunit
Rollout 89, Act: -2.730
33.333...ĠsoĠ2/5ĠequalsĠ66.666...ĊĊ
Rollout 71, Act: -2.816
ĠsuchĠthatĠtheyĠsumĠtoĠzero,ĠtheirĠpairwiseĠproductsĠsumĠtoĠ-2011,ĠandĠtheirĠproduct
Rollout 71, Act: -2.854
abc=Ġ-m.ĊĊSumĠofĠsquaresĠisĠΣa2=Ġ(Σa)^2Ġ-Ġ

DOWN_PROJ

Top 16 Positive Activations
Rollout 18, Act: 2.312
ĠsidesĠbyĠ-7:ĊĊ5bĠ+4cĠ=Ġ-3ĊĊSo,Ġ5bĠ+4
Rollout 12, Act: 2.117
Ġ+Ġ[Ġnu2ĠK(k)Ġ]Ġ/Ġ(Ġ(1Ġ-Ġnu2)ĠnuĠ)ĊĊSimplifyĠterms
Rollout 12, Act: 2.098
Ġ)Ġ+Ġ[Ġnu2ĠK(k)Ġ]Ġ/Ġ(Ġ(1Ġ-Ġnu2)ĠnuĠ)ĊĊSimplify
Rollout 2, Act: 2.067
8.ĊĊTherefore,ĠequationĠbecomes:ĊĊ(5mĠ+5)âī¡5Ġmod8ĊĊSubtractĠ
Rollout 18, Act: 2.048
,Ġn)ĠinĠZ2ĠsuchĠthatĠpmĠ+ĠqnĠ=Ġ0ĠforĠsomeĠintegersĠpĠandĠq
Rollout 6, Act: 2.046
textbf{NP}Ġ\).ĊĊ**Answer:**ĠĠĊ\(Ġ\textbf{P}Ġ=Ġ\text
Rollout 11, Act: 2.032
^{iθ},Ġc=1/(r2Ġe^{i2θ}).ĠSoĠaĠ=ĠbĠ=
Rollout 71, Act: 2.020
Ġ8044ĠareĠsayĠ89^2=7921,Ġ90
Rollout 17, Act: 2.007
3/(A·s2))Ġ/Ġm3Ġ)Ġ=Ġkg/(A·s2)Ġ=ĠT
Rollout 62, Act: 1.999
=(4)/2=2ĠorĠ(-2)/2=-1âī¡12Ġmod13.ĠSo
Rollout 89, Act: 1.975
,ĠmaybeĠ2/5thĠisĠ2ĠfifthsâĢĶWait,Ġunits.ĊĊWait,ĠmaybeĠit
Rollout 45, Act: 1.973
Therefore,ĊĊ2aĠ+ĠbĠ+Ġ3cĠ=Ġ14Ġ[EquationĠ2]ĊĊWait
Rollout 53, Act: 1.968
1.5.ĠMultiplyĠbothĠsidesĠbyĠ3:ĊĊsqrt(Ġc^2Ġ+Ġd^2Ġ)
Rollout 58, Act: 1.962
iĠ+1Ġ))Ġ*Ġ(i/(iĠ+1Ġ))^{iĠ-1}Ġ/Ġ(Ġ(i
Rollout 17, Act: 1.960
Ω·s,ĠbutĠperhapsĠbetterĠinĠbaseĠunits:ĠJĠ=Ġkg·m2/s2,Ġso
Rollout 98, Act: 1.956
ĠKĠorĠM-type,Ġright?ĠSoĠsomething'sĠwrongĠhere.ĠAĠstarsĠareĠhotterĠandĠmoreĠmassive.
Top 16 Negative Activations
Rollout 43, Act: -2.403
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊThereĠisĠaĠC-NOTĠgate
Rollout 47, Act: -2.403
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠtrainĠwithĠcross-sectionalĠareaĠ$
Rollout 55, Act: -2.403
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĊ\[\mathbf{
Rollout 63, Act: -2.403
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊLetĠĠ$P$ĠĠdenote
Rollout 60, Act: -2.403
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequency
Rollout 78, Act: -2.403
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>userĊJonĠandĠSteveĠrideĠtheirĠbicyclesĠalong
Rollout 55, Act: -2.405
ĠbecauseĠcos(ÏĢ/6)Ġ=ĠâĪļ3/2ĠandĠsin(ÏĢ/6)Ġ=
Rollout 30, Act: -2.406
Ġ[1/(a_kĠ-1Ġ)Ġ-1/a_kĠ]Ġ=Ġ1/(a_k(a_k
Rollout 30, Act: -2.416
n+1}Ġ-1/a_{n+3}).ĠThenĠthisĠwasĠequalĠtoĠ[b_{n
Rollout 30, Act: -2.420
1/a_{n+1}Ġ-Ġ1/a_{n+3})ĠasĠaĠdifferenceĠs_n
Rollout 30, Act: -2.477
Ġ(1/a_{n+1}Ġ-1/a_{n+3}).ĠThenĠthisĠwasĠequalĠto
Rollout 56, Act: -2.514
Ġ=Ġ|1*(âĪļ3Ġ-Ġ(-âĪļ3/2))Ġ+Ġ0*(-âĪļ3/2
Rollout 66, Act: -2.514
2Ġ+n2Ġ)Ġ-Ġ(m2Ġ-n2Ġ)Ġ)Ġ=Ġgcd(Ġm2Ġ+n
Rollout 91, Act: -2.600
Ġ[Ġ(tĠ+1Ġ)Ġ-Ġ(1Ġ-ĠtĠ)Ġ]Ġ/ĠtĠ=Ġ(ĠtĠ+1Ġ-
Rollout 95, Act: -2.701
12$Ġ.ĠPointsĠ$AĠ=ĠP_{0},ĠP_{1},ĠP_{2},Ġ...
Rollout 56, Act: -2.736
7/6*(âĪļ3Ġ-Ġ(-âĪļ3/2Ġ))|Ġ/Ġ2ĊĊSimplifyĠterm

Layer 38

GATE_PROJ

Top 16 Positive Activations
Rollout 77, Act: 3.415
,52)=364.ĊĊ**FinalĠAnswer**Ċ\boxed{364}ĊĊ<|im_start|>answer
Rollout 36, Act: 3.381
:Ġ\boxed{192}ĊĊ**FinalĠAnswer**Ċ\boxed{192}ĊĊ<|im_start|>answer
Rollout 58, Act: 3.372
037171}.ĊĊ**FinalĠAnswer**Ċ\boxed{203717
Rollout 60, Act: 3.355
ĠtheĠanswerĠshouldĠbeĠ6ĠHz.ĊĊ**FinalĠAnswer**Ċ\boxed{6}ĊĊ<|im_start|>answerĊA
Rollout 67, Act: 3.346
ĠasĠconcluded.ĠTherefore,ĠcorrectĠanswer.ĊĊ**FinalĠAnswer**Ċ\boxed{348}ĊĊ<|im_start|>answer
Rollout 63, Act: 3.333
Ġf(p,q)Ġ=ĠpqĠ\).ĊĊ**FinalĠAnswer**Ċ\boxed{f(p,q)Ġ=Ġpq
Rollout 68, Act: 3.329
ĠtheĠdesiredĠanswerĠis501.ĊĊ**FinalĠAnswer**Ċ\boxed{501}ĊĊ<|im_start|>answer
Rollout 38, Act: 3.324
ĠisĠ\boxed{61}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{61}ĊĊ<|im_start|>answerĊ
Rollout 62, Act: 3.299
Ġ\boxed{132}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{132}ĊĊ<|im_start|>answer
Rollout 56, Act: 3.296
Ġsqrt3.ĠTherefore,ĠweĠconclude.ĊĊ**FinalĠAnswer**Ċ\boxed{\sqrt{3}}ĊĊ<|im_start|>answer
Rollout 93, Act: 3.292
4}}}ĠsinceĠtheĠclearestĠexpression.ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{100
Rollout 59, Act: 3.289
ĠtheĠanswerĠisĠ515.ĊĊ**FinalĠAnswer**Ċ\boxed{515}ĊĊ<|im_start|>answer
Rollout 66, Act: 3.287
ĠcorrectĠanswerĠisĠ182.ĊĊ**FinalĠAnswer**Ċ\boxed{182}ĊĊ<|im_start|>answer
Rollout 70, Act: 3.283
\boxed{315}.ĊĊ**FinalĠAnswer**Ċ\boxed{315}ĊĊ<|im_start|>answer
Rollout 16, Act: 3.281
53.ĊĊTherefore,ĠboxĠthat.ĊĊ**FinalĠAnswer**Ċ\boxed{789953
Rollout 71, Act: 3.276
ĠisĠ\boxed{98}.ĊĊ**FinalĠAnswer**Ċ\boxed{98}ĊĊ<|im_start|>answerĊ
Top 16 Negative Activations
Rollout 31, Act: -0.824
1Ġ-âĪļ5)^2]/4=(1Ġ-2âĪļ5Ġ+5)/4=(6Ġ-
Rollout 49, Act: -0.824
ĠaĠÏĥ-compactĠsubspaceĠisĠaĠcountableĠunionĠofĠclosedĠ(compact)Ġsets.ĊĊButĠhowĠdoes
Rollout 40, Act: -0.831
Ġ+ĠâĪļ2)Ġ=Ġ4Ġ-Ġ2Ġ=Ġ2ĊĊNumerator:ĊĊ[Ġ(1
Rollout 87, Act: -0.842
.5):Ġr=sqrt(0.25+0.25+0.25
Rollout 99, Act: -0.885
3362240124Ġ-2.65731324
Rollout 17, Act: -0.899
=1;Ġθ=ÏĢ,Ġu=-1.ĠSoĠintegralĠbecomes:ĊĊâĪ«_{-1}
Rollout 4, Act: -0.903
{eq}}Ġ\)Ġis:ĊĊ\[Ċ\frac{\DeltaĠC_{\text{eq}}}{
Rollout 89, Act: -0.907
500/3ĠâīĪ200/3âīĪ66.666...ĊĊCannot
Rollout 40, Act: -0.935
)(2Ġ+ĠâĪļ2)Ġ=Ġ4Ġ-Ġ2Ġ=Ġ2ĊĊNumerator:ĊĊ[
Rollout 70, Act: -0.958
0.3567ĊĊSo,Ġ(-1.2528)/(0.35
Rollout 49, Act: -1.004
ĠofĠX.ĠThenĠXâĢĻâĢĻĠisĠtheĠsetĠofĠlimitĠpointsĠofĠXâĢĻ,ĠandĠsoĠon.ĠIf
Rollout 31, Act: -1.006
2*(1Ġ-âĪļ5)/2=(1Ġ-5)/4=Ġ(-4)/4=Ġ-
Rollout 42, Act: -1.040
ĠtoĠformĠH2O+,ĠwhichĠleavesĠtoĠformĠaĠtertiaryĠcarbocationĠatĠtheĠcentralĠcarbon.ĠNow,
Rollout 12, Act: -1.059
Ġf(\nu)=Ġ\int_1^{\frac{1}{\nu}}Ġ\frac{dx
Rollout 75, Act: -1.079
ĠtheĠresultingĠ2k=Ġ(SĠ-3)/4=2*(a2Ġ+b2Ġ+
Rollout 71, Act: -1.087
12d2Ġ-12dĠâī¡1Ġ-0âī¡1Ġmod4.ĠTherefore,Ġk

UP_PROJ

Top 16 Positive Activations
Rollout 1, Act: 2.398
)/2Ġ=Ġ(3d2Ġ/Ġ2)/2Ġ=Ġ3d2/4.ĊĊãĢĪ
Rollout 83, Act: 2.306
!ĠforĠeachĠgroup'sĠpermutations)Ġ=>Ġ3!Ġ*Ġ(3!Ġ*Ġ3!Ġ*Ġ
Rollout 11, Act: 2.282
Ġ).ĠThus,Ġp=Ġ(t1Ġ+1/t1Ġ)Ġ+Ġ(t2Ġ+1/t2
Rollout 38, Act: 2.268
ĠisĠx/2âĩĴx=2*(x/2)=x.ĠSoĠno,Ġit'sĠL
Rollout 17, Act: 2.252
/AĠ*ĠJ/TĠ=Ġ(T·m/A)Ġ*Ġ(J/T)Ġ=Ġ(m/A)*(
Rollout 11, Act: 2.233
Ġ=Ġ\overline{a}Ġ+Ġ\overline{b}Ġ+Ġ\overline{c}
Rollout 28, Act: 2.232
Ġ=Ġ-BĠkĠ/Ġ(2hĠ-a)Ġ...Ġ(1a)ĊĊFromĠequationĠ(2):ĊĊ
Rollout 49, Act: 2.220
2^{|X|Ġ*Ġ\aleph_0}Ġ=Ġ2^{|X|}.ĠBut
Rollout 12, Act: 2.216
ComputeĠ1Ġ-Ġ(t2Ġ-Ġν2)/(1Ġ-Ġν2)Ġ)ĊĊ=Ġ(1Ġ-
Rollout 11, Act: 2.197
.ĠLet'sĠwriteĠcĠasĠ1/(ab).ĠThenĠqĠbecomesĠa/bĠ+Ġb/(1/(
Rollout 77, Act: 2.188
3+5)*15=8*15=120ĊĊRight-handĠside:5
Rollout 11, Act: 2.183
ab).ĠSoĊĊ\overline{aĠb}Ġ=Ġ\frac{1}{aĠb}ĠâĨĴ
Rollout 69, Act: 2.181
i},Ġ3Ġtheta_iĠ=30°Ġ/3^{i}=Ġtheta_{iĠ-1}\
Rollout 11, Act: 2.180
a*bĠ)Ġ=1/(a^2Ġz).ĠHence,ĠcĠ=1/(a^2Ġz
Rollout 11, Act: 2.165
ĠholdĠifĠabcĠ=Ġ1/(aĠbĠcĠ),ĠimplyingĠ(abc)^2Ġ=Ġ1,Ġso
Rollout 85, Act: 2.165
(2,0)=4.ĠSoĠsetĠ{2,3,4}Ġâľĵ.ĊĊCheckĠ(x
Top 16 Negative Activations
Rollout 5, Act: -2.169
181.25Ċ\]ĊTheĠgreatestĠintegerĠnotĠexceedingĠ181.25
Rollout 50, Act: -2.174
\[Ċ\boxed{112}Ċ\]<|im_end|>
Rollout 61, Act: -2.177
,ĠtheĠfractionĠisĠinĠsimplestĠform.ĠTherefore,Ġ\(ĠpĠ=Ġ133Ġ\)ĠandĠ\(
Rollout 32, Act: -2.186
Ġ+Ġ\mathbf{M}^2Ġ+Ġ\mathbf{M}^3Ġ+Ġ\dots
Rollout 41, Act: -2.187
927\)ĊĊTheĠnumberĠofĠpositiveĠintegersĠlessĠthanĠ10,000ĠthatĠhave
Rollout 45, Act: -2.201
Ġ\boxed{-1}.ĠHowever,ĠtheĠquestionĠisĠtheĠgreatestĠpossibleĠvalueĠofĠm,ĠsoĠ(b)
Rollout 73, Act: -2.203
398.ĠHence,ĠtheĠrequiredĠremainderĠisĠ398.ĠTherefore,ĠtheĠanswerĠisĠ
Rollout 95, Act: -2.208
(\overline{P_kQ_k}\)ĠwithĠrationalĠlengths.ĊĊ1.Ġ**CoordinatesĠandĠSimilarĠTri
Rollout 21, Act: -2.216
Ġ\)ĠandĠ\(ĠSĠ\subseteqĠXĠ\).ĊĊ###ĠFinalĠAnswersĊĊ(a)Ġ\boxed{SĠ\
Rollout 22, Act: -2.218
m/s}Ġ\)ĊĊ**FinalĠAnswer**:Ċ\[Ċ\boxed{1694}Ġ\
Rollout 13, Act: -2.219
12Ċ\]ĊĊ\[Ċ\boxed{187.12}Ċ\]<|im_end|>
Rollout 9, Act: -2.230
ĠareaĠcommonĠtoĠtriangleĠ\(GEM\)ĠandĠsquareĠ\(AIME\)ĠisĠ80ĠsquareĠunits.
Rollout 41, Act: -2.232
927.ĊĊ**FinalĠAnswer**ĊTheĠnumberĠofĠpositiveĠintegersĠlessĠthanĠ10,00
Rollout 41, Act: -2.254
ĠthanĠ10,000ĠthatĠhaveĠatĠmostĠtwoĠdifferentĠdigitsĠisĠ\(\boxed{9
Rollout 44, Act: -2.296
Ġpositive.ĠStatementĠ[D]ĠisĠincorrect.ĊĊFinalĠAnswer:Ġ\(\boxed{ABC}\)<|im_end|>
Rollout 82, Act: -2.309
{2}Ċ\]ĊĊTheĠcodewordsĠare:Ċ-Ġ\(Ġ\frac{16}{69

DOWN_PROJ

Top 16 Positive Activations
Rollout 91, Act: 2.523
Ġ(tĠ-1Ġ)*0Ġ-Ġ(-tĠ)*(-2Ġ)Ġ|Ġ=Ġ|0Ġ-2tĠ|=
Rollout 28, Act: 2.498
/(2kĠ-b))Ġ*bĠ)Ġ/aĠ)ĊĊLetĠmeĠwriteĠBĠonĠbothĠsidesĠ(assuming
Rollout 91, Act: 2.449
(tĠ-1Ġ)*0Ġ-Ġ(-tĠ)*(-2Ġ)Ġ|=Ġ|-2tĠ|Ġ=2t
Rollout 31, Act: 2.372
0ĊĊSinceĠthisĠmustĠholdĠforĠallĠpossibleĠa0ĠandĠa1,ĠwhichĠmeansĠtheĠcoefficientsĠofĠa
Rollout 81, Act: 2.320
ĠAB:Ġm1=Ġ(0Ġ-h)/(-17Ġ-0)=h/17.ĊĊ
Rollout 83, Act: 2.318
ĠtheĠsecondĠgivesĠ(r_{i+1}Ġ+Ġr_{i+2}Ġ+Ġr_{i
Rollout 64, Act: 2.231
Ġ+2*(Ġ(a2Ġ-a1)/2Ġ)Ġ=2a1Ġ-a2Ġ+a
Rollout 7, Act: 2.220
Ġ5-pointĠincrease,ĠeveryĠparticipant'sĠscoreĠisĠincreasedĠbyĠ5.ĠTherefore,ĠtheĠnewĠtotalĠscore
Rollout 56, Act: 2.205
ÎĶy=Ġ(11âĪļ3)/sqrt(559)Ġ*Ġsqrt(559
Rollout 1, Act: 2.203
Ġ/Ġ2)/2Ġ=Ġ3d2/4.ĊĊãĢĪy,ĠyãĢīĠ=Ġ||y
Rollout 30, Act: 2.193
Ġb_nĠ]Ġ/Ġ[a_{n+1}(a_{n+1}Ġ-1Ġ)Ġ]
Rollout 75, Act: 2.187
3ĠmodĠ4.ĠAlternatively,ĠifĠweĠhaveĠsevenĠsquaresĠ(ifĠweĠneededĠseven),ĠbutĠhereĠit
Rollout 67, Act: 2.180
.ĊĊLetĠmeĠdenoteĠtheĠgeneralĠtermĠc_nĠasĠAn2Ġ+ĠBnĠ+ĠC.ĠThenĠgivenĠthat
Rollout 96, Act: 2.177
69Ġ=27Ġ-Ġ(-12)*(-526)/169ĊĊComputeĠx
Rollout 96, Act: 2.141
697*895:ĊĊComputeĠ700*895=626
Rollout 46, Act: 2.140
Ġ0)ĠandĠM2ĠisĠatĠpositionĠ(-b,Ġ0)ĠinĠtheĠrotatingĠframe.ĠBut
Top 16 Negative Activations
Rollout 10, Act: -2.098
ĠofĠnumbersĠn_iĠwhereĠeachĠn_iĠisĠaĠlosingĠposition,ĠandĠthenĠshowĠifĠweĠcanĠconstructĠn
Rollout 28, Act: -2.103
ĠRemember,ĠaĠhyperbolaĠhasĠ\(ĠB^2Ġ-Ġ4ACĠ>Ġ0Ġ\).ĊĊFrom
Rollout 10, Act: -2.113
,1,2,3,...ĊĊTherefore,ĠtheĠlosingĠpositionsĠareĠnumbersĠofĠtheĠformĠ5kĠand
Rollout 49, Act: -2.119
Ġcontinuum.ĠTheĠoriginalĠproblemĠstatesĠthatĠXĠmustĠbeĠcountable.ĊĊSoĠtoĠbridgeĠtheĠgapĠbetweenĠ2
Rollout 82, Act: -2.121
Ġ8s.ĊĊWait,ĠmoreĠprecisely:ĊĊWait,ĠinitialĠprobabilities:12,10,8
Rollout 66, Act: -2.176
.ĠSinceĠinĠaĠright-angledĠtriangle,ĠtheĠaltitudeĠprojectsĠtheĠhypotenuseĠintoĠsegmentsĠproportionalĠtoĠthe
Rollout 93, Act: -2.186
ime,ĠthenĠyes,ĠbutĠkĠrangesĠfromĠ1ĠtoĠ1007,ĠandĠasĠ
Rollout 10, Act: -2.199
uentĠtoĠ0ĠorĠ2Ġmodulo5ĠareĠlosingĠpositions.ĠNow,ĠhowĠtoĠcheckĠifĠthisĠpattern
Rollout 66, Act: -2.203
GivenĠthatĠinĠaĠright-angledĠtriangle,ĠtheĠaltitudeĠtoĠtheĠhypotenuseĠisĠhĠ=Ġ(leg
Rollout 10, Act: -2.219
7âī¡2Ġmod5ĊĊSoĠactually,ĠtheĠlosingĠpositionsĠareĠnumbersĠcongruentĠtoĠ0ĠorĠ
Rollout 69, Act: -2.246
)/cosθĠcos3θĠ]Ġ=Ġsin2ĠthetaĠ/Ġco[s3Ġtheta,ĊĊSoĠalphaĠ(cos
Rollout 38, Act: -2.257
r2Ġ-625rĠ+168=0.ĊĊUsingĠquadraticĠformula:ĊĊr=(6
Rollout 93, Act: -2.282
2Ġ+ĠC(2014,1007}=ĠifĠthere'sĠaĠidentityĠasĠ
Rollout 38, Act: -2.382
2Ġ-1250ĠrĠ+336Ġ=0ĊĊLetâĢĻsĠsolveĠforĠrĠwithĠquadratic
Rollout 66, Act: -2.876
ĠrecallĠthatĠinĠaĠright-angledĠtriangle,ĠtheĠaltitudeĠtoĠtheĠhypotenuseĠhasĠaĠlengthĠequalĠto
Rollout 62, Act: -4.277
ĠmĠism=132,ĠwhichĠ11^3*13.ĠWhichĠhasĠfour

Layer 39

GATE_PROJ

Top 16 Positive Activations
Rollout 46, Act: 1.251
-\frac{GĠM_{2}}{\rho_{2}}-\frac{\omega^{2}}{
Rollout 21, Act: 1.230
ĠĠwhichĠisĠaĠfiniteĠnonemptyĠstringĠformedĠfromĠaĠsetĠĠ$x_1,Ġx_2
Rollout 40, Act: 1.212
Ġcoordinates,Ġ|p|Ġ+Ġ|q|.ĊĊHmm,Ġokay.ĠLet'sĠbreakĠdownĠwhatĠeachĠmove
Rollout 60, Act: 1.141
'sĠaboutĠtheĠDopplerĠeffectĠandĠbeatĠfrequency.ĠHmm,Ġlet'sĠstartĠbyĠunderstandingĠtheĠquestion.ĊĊSo
Rollout 3, Act: 1.125
ĠcalculateĠtheĠmeanĠmolecularĠspeedĠofĠradonĠgas.ĠHmm,Ġfirst,ĠIĠrememberĠthere'sĠaĠformulaĠfor
Rollout 99, Act: 1.106
ĠneedĠtoĠfindĠtheĠdecimalĠvalueĠofĠx3.ĠHmm,Ġokay.ĠĊĊFirst,ĠIĠrememberĠthatĠthe
Rollout 91, Act: 1.076
ĠtheĠvalueĠofĠ|cotĠBĠ-ĠcotĠC|.ĠHmm,Ġright.ĠOkay,Ġfirst,ĠletĠmeĠvisualize
Rollout 52, Act: 1.062
.ĠWeĠneedĠtoĠfindĠtheĠsmallestĠyummyĠinteger.ĠHmm,Ġinteresting.ĠSoĠIĠneedĠtoĠfigureĠoutĠwhat
Rollout 48, Act: 1.054
00Ġ008BĠ5BC6ĠAEE0Ġ.ĠDoesĠtheĠheaderĠcontainsĠerror?Ċ
Rollout 60, Act: 1.043
.ĠTheĠsoundĠreflectsĠoffĠtheĠcarĠandĠcomesĠbackĠtoĠtheĠsource,ĠwhereĠit'sĠsuperimposedĠwith
Rollout 42, Act: 1.040
ĠproducedĠwhenĠcyclobutyl(cyclopropyl)methanolĠreactsĠwithĠphosphoricĠacidĠinĠwater
Rollout 93, Act: 1.037
ĠofĠ(ÏĢk/1007).ĠHmm.ĠLetĠmeĠwriteĠthatĠdownĠtoĠmakeĠitĠclearer
Rollout 87, Act: 1.035
ĠdensityĠrho,ĠandĠeachĠsideĠisĠlengthĠa.ĠHmm.ĠLetĠmeĠstartĠbyĠrecallingĠsomeĠbasicĠconceptsĠabout
Rollout 48, Act: 1.028
0Ġ008BĠ5BC6ĠAEE0Ġ.ĠDoesĠtheĠheaderĠcontainsĠerror?Ċ<|im_start|>
Rollout 97, Act: 1.021
mathrm{O}_{2}(g)$ĠisĠstartedĠinĠaĠclosedĠcylinderĠunderĠisothermalĠisochoricĠcondition
Rollout 72, Act: 1.013
Ġ\(ĠnĠ\)ĠwhoseĠsumĠisĠ1.ĠHmm,ĠrootsĠofĠunity...soĠtheseĠareĠcomplexĠnumbers
Top 16 Negative Activations
Rollout 22, Act: -3.418
0).ĠĊĊSinceĠ1,6942Ġ=Ġ2,869,63
Rollout 73, Act: -3.421
ĠisĠindeedĠ398.ĠLet'sĠcomputeĠ16398Ġ-Ġ16*
Rollout 3, Act: -3.430
ĠKĠ(unlessĠstatedĠotherwise),ĠperhapsĠanswerĠaroundĠ\boxed{169}Ġm/s.ĠButĠneed
Rollout 71, Act: -3.449
ĠIsĠ1417ĠaĠperfectĠsquare?Ġ372=1369,Ġ
Rollout 16, Act: -3.452
096ĠdividedĠbyĠ217.Ġ217*5=1085
Rollout 3, Act: -3.475
50)=161.4,ĠasĠ161^2=25,9
Rollout 96, Act: -3.510
8Ġ÷2197=3ĠsinceĠ2197*3=659
Rollout 81, Act: -3.513
4,721=0.ĊĊUseĠquadraticĠformula:ĊĊxĠ=Ġ[17,242
Rollout 89, Act: -3.548
/3ĠisĠacceptable.ĊĊThereforeĠwrittenĠboxedĠasĠ\boxed{\dfrac{200}{3}}
Rollout 3, Act: -3.597
Ġm/s.ĠThus,ĠpresentĠanswerĠboxedĠasĠ\boxed{169Ġ\text{Ġm/s}}
Rollout 22, Act: -3.641
).ĠĊĊSinceĠ1,6942Ġ=Ġ2,869,636
Rollout 26, Act: -3.653
,ĠsoĠtwoĠdecimalĠplaces,ĠtheĠanswerĠisĠ\boxed{90.39}ĊĊĊ**FinalĠAnswer
Rollout 90, Act: -3.657
8:ĠLet'sĠcomputeĠ2018ÃĹ50,000=100
Rollout 60, Act: -3.689
ĠdividedĠbyĠ82ĠisĠ6,ĠsinceĠ82*6Ġ=Ġ492,
Rollout 3, Act: -3.829
ĠtheĠmeanĠmolecularĠspeedĠofĠradonĠgasĠisĠ\boxed{170}Ġm/sĠifĠweĠround
Rollout 3, Act: -3.965
ĠthisĠquestionĠisĠambiguous.ĠNevertheless,ĠlikelyĠanswerĠ\boxed{170}Ġm/s.ĠAlternatively,

UP_PROJ

Top 16 Positive Activations
Rollout 59, Act: 2.771
ĠnĠ=Ġ123Ġ+Ġ392Ġ=Ġ515\).ĊĊ\[Ċ\
Rollout 39, Act: 2.755
NĠ=Ġ\frac{10^{322}Ġ-Ġ10}{9}Ġ-Ġ
Rollout 41, Act: 2.751
9Ġ+Ġ90Ġ+Ġ252Ġ+Ġ576Ġ=Ġ927\
Rollout 68, Act: 2.741
Ġ\):ĠIntegersĠfromĠ\(-500\)ĠtoĠ\(-1\)ĠgiveĠ500
Rollout 40, Act: 2.732
5iĠ+Ġ5(1Ġ+Ġi(\sqrt{2}Ġ+Ġ1))(1Ġ+Ġi
Rollout 84, Act: 2.725
)Ċ\]ĊĊExpandingĠandĠsimplifying:ĊĊ\[Ċ9Ġ-Ġ36xĠ=Ġ4Ġ-
Rollout 8, Act: 2.722
Ġ\(240Ġ+Ġ7200Ġ=Ġ7440\).ĊĊTheĠprobabilityĠis
Rollout 61, Act: 2.717
Ġ+Ġ13Ġ+Ġ7Ġ+Ġ1Ġ=Ġ133Ġ\).ĊĊThus,ĠtheĠprobability
Rollout 58, Act: 2.711
).ĊĠĠĠ-ĠSimplifying,ĠeachĠtermĠforĠpartĠ\(ĠiĠ\)ĠreducesĠtoĠ\(ĠP(0)
Rollout 32, Act: 2.685
669Ġ\cdotĠ6Ġ+Ġ4Ġ=Ġ4014Ġ+Ġ4Ġ=Ġ
Rollout 41, Act: 2.679
9Ġ+Ġ216Ġ+Ġ27Ġ=Ġ252\)ĊĊ4.Ġ**4
Rollout 8, Act: 2.674
2400Ġ+Ġ1200Ġ=Ġ7200\).ĊĊAddingĠbothĠcases
Rollout 61, Act: 2.671
Ġ1Ġ\)ĊĊSummingĠthese,ĠweĠgetĠ\(Ġ37Ġ+Ġ31Ġ+Ġ2
Rollout 64, Act: 2.671
}^8Ġ\left(2^{2012}Ġ-Ġk(2^{200
Rollout 41, Act: 2.653
\)ĊĊTotal:Ġ\(9Ġ+Ġ90Ġ+Ġ252Ġ+Ġ576Ġ=
Rollout 20, Act: 2.653
First,ĠfactorizeĠ2021:Ġ\(Ġ2021Ġ=Ġ43Ġ\
Top 16 Negative Activations
Rollout 84, Act: -1.994
2).ĠWait,ĠbutĠthat'sĠdifferent.ĊĊWait,ĠdependingĠonĠhowĠyouĠreadĠit:Ġ"twice
Rollout 96, Act: -1.997
Ġsystem,ĠpointĠAĠisĠonĠtheĠpositiveĠx-axis.ĠWait,Ġactually,ĠinĠtheĠnewĠsystem,Ġsince
Rollout 16, Act: -1.999
4Ġdays.ĠThatĠseemsĠlow.ĠHow?ĊĊWait,ĠbutĠinĠourĠproblem,Ġit'sĠaĠbitĠdifferent
Rollout 8, Act: -2.005
,ĠexamplesĠincludeĠRRRRRĠandĠGGGGG.ĊĊSoĠactually,ĠifĠallĠredĠcardsĠinĠtheĠlayout
Rollout 23, Act: -2.010
ĠweĠareĠsupposedĠtoĠuseĠtheĠ"partialĠhalf-life"ĠconceptĠwhereĠtheĠdecayĠconstantsĠareĠconsideredĠperĠdecayĠmode
Rollout 56, Act: -2.012
ĠthenĠthatĠtriangleĠisĠreplacingĠorĠbeingĠadded?ĠWait,ĠtheĠproblemĠsaysĠ"theĠnewĠpastureĠisĠtriangleĠA
Rollout 97, Act: -2.021
ĠamountĠinĠmoles.ĠItĠjustĠgivesĠaĠpressure.ĠButĠsinceĠit'sĠclosedĠandĠunderĠconstantĠvolumeĠand
Rollout 89, Act: -2.045
5ĠofĠtheĠnumberĠofĠboys.ĠHmm.ĠWait,Ġlet'sĠcheckĠagain.ĊĊTheĠproblemĠsays:Ġ"
Rollout 56, Act: -2.061
ĠA3,ĠsuchĠthatĠA3A0ĠisĠhalfĠofĠA2A0.ĠSoĠtheĠoriginalĠA
Rollout 47, Act: -2.097
ĠtunnelĠwallsĠareĠmoving.ĠHmm,ĠperhapsĠboundaryĠconditionsĠcomplicateĠthingsĠaĠbitĠhere.ĠMaybeĠweĠcanĠassume
Rollout 47, Act: -2.116
ĠtheĠairĠthat'sĠmovingĠwithĠrespectĠtoĠtheĠtunnel?ĠHmm.ĊĊThisĠmightĠbeĠgettingĠconfusing.ĠLet's
Rollout 47, Act: -2.118
ĠmovingĠtowardsĠtheĠtrainĠatĠspeedĠvt?ĠOrĠisĠitĠtheĠairĠthat'sĠmovingĠwithĠrespectĠtoĠtheĠtunnel
Rollout 83, Act: -2.173
16;Ġ1296?ĠWait,Ġwait.ĠWait.ĊĊWait.ĊĊWaitĠourĠmistakeĠis
Rollout 96, Act: -2.229
ĠIĠfoundĠareĠnotĠtheĠactualĠcoordinatesĠinĠtheĠnewĠsystem.ĠMaybeĠsomething'sĠwrongĠinĠtheĠparametrization
Rollout 84, Act: -2.364
Ġdrank?ĠOrĠtwiceĠtheĠoriginalĠpurchasedĠamount?ĠHmm,ĠtheĠproblemĠsaysĠ"twiceĠasĠmuch".ĠWhen
Rollout 77, Act: -2.436
,Ġ4)ĠshouldĠbeĠ4?ĠWait,ĠinĠmyĠpreviousĠexample,ĠIĠcomputedĠf(2

DOWN_PROJ

Top 16 Positive Activations
Rollout 41, Act: 2.314
3,4),ĠnumbersĠofĠlengthĠnĠusingĠexactlyĠtwoĠdifferentĠdigits.ĠForĠn=1,ĠobviouslyĠzero
Rollout 37, Act: 2.283
ĠentriesĠinĠaĠrowĠareĠtheĠsame).ĠButĠaĠrankĠ1ĠmatrixĠhasĠdeterminantĠzero,ĠbecauseĠallĠrows
Rollout 1, Act: 2.255
ĠdĠ/Ġsqrt(2)ĠtimesĠsqrt(2)ĠfromĠcenterĠy,ĠwhichĠisĠexactlyĠd.ĠWait
Rollout 37, Act: 2.217
ĠinĠaĠrowĠareĠtheĠsame).ĠButĠaĠrankĠ1ĠmatrixĠhasĠdeterminantĠzero,ĠbecauseĠallĠrowsĠare
Rollout 20, Act: 2.179
ĠsimilarĠstepsĠasĠbefore:ĠmultiplyingĠbyĠ(p-1),ĠbringingĠp^{knĠ+1}Ġ-1
Rollout 17, Act: 2.176
m2/s2,ĠsoĠ(kg·m2/s2Ġ*m)/AĠ=Ġkg·m
Rollout 16, Act: 2.175
ĠofĠnĠgeometric(p)ĠvariablesĠisĠequalĠtoĠsum_{k=1}^{infty}Ġ[1
Rollout 37, Act: 2.170
ĠareĠtheĠsame).ĠButĠaĠrankĠ1ĠmatrixĠhasĠdeterminantĠzero,ĠbecauseĠallĠrowsĠareĠmultiplesĠofĠeach
Rollout 49, Act: 2.163
izableĠspaceĠisĠLindelöfĠandĠmetrizable,ĠhenceĠsecondĠcountable).ĠButĠ2nd
Rollout 15, Act: 2.161
Ġ0.ĊĊThisĠisĠequivalentĠtoĠx_{t+1}[i]Ġ=Ġ(x_{t}[
Rollout 97, Act: 2.143
t,ĠintegrateĠtoĠin(-2k)Pt}ĊĊThusĠPressureĠisĠP=ĠP0Ġe^{-2
Rollout 97, Act: 2.117
.ĠTheĠrateĠconstantĠ\(k\)ĠisĠ\(5Ġ\timesĠ10^{-4}Ġ\,Ġ\
Rollout 20, Act: 2.104
Ġm.ĠThenĠmultiplyingĠbothĠsidesĠbyĠ(pĠ-1):Ġp^{knĠ+1}Ġ-Ġ1
Rollout 29, Act: 2.102
ĠcontainerĠopenĠtoĠtheĠatmosphere,ĠtheĠgaugeĠpressureĠatĠtheĠbottomĠwouldĠjustĠbeĠrho*g*h,ĠwithĠh
Rollout 27, Act: 2.090
ĠrmsĠvelocity:ĊĠĠĠĠĠ\[ĊĠĠĠĠĠv_{z,Ġ\text{rms}}Ġ=Ġ\sqrt{\frac
Rollout 78, Act: 2.090
ĠgoingĠeastĠatĠ20Ġmph,ĠSteveĠisĠgoingĠwestĠatĠ20Ġmph.ĠThereĠareĠtwo
Top 16 Negative Activations
Rollout 17, Act: -2.126
ĠWait,Ġ1.41*8=11.28.ĠSoĠ8ÏĢĠ*
Rollout 19, Act: -2.145
ĠbutĠ>=6?Ġ83ĠitselfĠisĠprime,ĠsoĠonlyĠ1Ġand83.ĠSoĠyes
Rollout 87, Act: -2.150
ĠbetweenĠÏĢ/4ĠandĠÏĢ/2,ĠrĠgoesĠfromĠ0ĠtoĠ1/sinθ.
Rollout 22, Act: -2.181
ButĠhere'sĠanotherĠproblem.ĠSinceĠtheĠproblemĠisĠinĠhydrogenĠgas,ĠwhichĠinĠrealityĠtheĠmoleculesĠareĠdi
Rollout 95, Act: -2.194
ĠatĠk-thĠdivisionĠalongĠABĠfromĠAĠtoĠB,ĠdividedĠintoĠ2450ĠequalĠparts.
Rollout 22, Act: -2.204
Ġ2,800ĠâīĪĠ5,910,800.ĊĊDifference:
Rollout 26, Act: -2.206
0.39ĠkJ/mol,ĠandĠtheĠabsoluteĠvalueĠ|X|ĠwouldĠbeĠ90.
Rollout 3, Act: -2.227
/337âīĪ0.047.ĠHence,Ġ168.5Ġ+0
Rollout 40, Act: -2.247
2Ġ=ĠiâĪļ2Ġ+Ġi*(2)/2Ġ=ĠiâĪļ2Ġ+Ġi1ĊĊTherefore,
Rollout 26, Act: -2.253
)Ġ*ĠRĠ*ĠTĊĊButĠRĠisĠgivenĠasĠ8.3ĠJ/K/mol,ĠwhichĠis
Rollout 83, Act: -2.307
âĢĻsĠoriginalĠresidueĊĊGroup1âĢĻsĠresidueĠ=ĠGroup2âĢĻsĠoriginalĠresidueĊĊGroup2âĢĻsĠresidueĠ=ĠGroup
Rollout 83, Act: -2.310
ĠtoĠGroup0.ĠSimilarly,ĠrotationĠbyĠ6ĠpositionsĠwouldĠcycleĠtheĠgroupsĠinĠtheĠreverseĠorder.ĊĊMeanwhile
Rollout 16, Act: -2.334
953-1904=49.ĊĊgcd(68,49).ĊĊ6
Rollout 69, Act: -2.372
,Ġi=2:Ġ30°Ġ/9,Ġthat'sĠ10/3°,
Rollout 83, Act: -2.503
ĠGroup0.ĠSimilarly,ĠrotationĠbyĠ6ĠpositionsĠwouldĠcycleĠtheĠgroupsĠinĠtheĠreverseĠorder.ĊĊMeanwhile,
Rollout 66, Act: -2.579
ĠPythagoreanĠtriple.ĠTheĠsidesĠareĠthusĠofĠtheĠformĠm2Ġ-Ġn2,Ġ2mn

Layer 40

GATE_PROJ

Top 16 Positive Activations
Rollout 53, Act: 3.467
:ĊĊ1971Ġ÷146:ĊĊLetĠmeĠfactorĠnumeratorĠandĠdenominator:ĊĊ197
Rollout 13, Act: 3.289
351936.ĊĊBreakingĠitĠdown:ĊĊ6000*1Ġ=Ġ60
Rollout 58, Act: 3.284
8Ġ*Ġ2019:ĊĊFirst,Ġ2018Ġ*Ġ2000
Rollout 53, Act: 3.264
Ġ1971Ġ÷146:ĊĊ146*13=189
Rollout 96, Act: 3.264
06*169=ĠLet'sĠcompute:ĊĊ4106*100=4
Rollout 22, Act: 3.257
6e23Ġkg/moleculeĊĊCalculateĠthat:ĊĊ0.002016Ġ/Ġ
Rollout 96, Act: 3.252
ĠcheckĠ1740Ġ÷13:ĊĊ13*133Ġ=Ġ17
Rollout 13, Act: 3.249
36ĊĊCalculatingĠthatĠvalue:ĊĊFirst,Ġ1Ġ*Ġ6000Ġ=Ġ6
Rollout 20, Act: 3.245
*43*47.ĊĊCalculatingĠthat:ĊĊMultiplyĠstepĠbyĠstep:ĊĊ2*3=6
Rollout 53, Act: 3.223
1971ĠdividedĠbyĠ146:ĊĊ1971Ġ÷146:ĊĊ
Rollout 13, Act: 3.212
51936ĊĊCalculatingĠthatĠvalue:ĊĊFirst,Ġ1Ġ*Ġ6000
Rollout 31, Act: 3.210
Ġ(xĠ+1).ĠLet'sĠdoĠpolynomialĠdivision:ĊĊx^3Ġ-0x^2Ġ-2
Rollout 4, Act: 3.210
,150ĠbyĠ5025:ĊĊFirst,Ġ5025Ġ*Ġ1
Rollout 23, Act: 3.209
10Ġ*945âīĪĠLet'sĠsee:Ġ945Ġ*0.2=1
Rollout 61, Act: 3.208
1.ĠLet'sĠsumĠthat.ĠLet'sĠcompute:ĊĊ(37Ġ+31)Ġ=68
Rollout 60, Act: 3.196
95Ġ*Ġ492ĊĊFirst,Ġ492Ġ*Ġ1Ġ=Ġ49
Top 16 Negative Activations
Rollout 87, Act: -1.054
:ĊĊ\[ĊVĠ=Ġ\frac{\rhoĠa^2}{4\pi\epsilon_0}
Rollout 96, Act: -1.065
theta\)ĠisĠtheĠangleĠlineĠ\(L\)ĠmakesĠwithĠtheĠoriginalĠx-axis.ĠTheĠrotationĠmatrixĠforĠ\
Rollout 44, Act: -1.070
ĠisĠexpandedĠreversiblyĠwithĠ$\mathrm{T}_{1}=\mathrm{T}_{2}$,ĠandĠ(ii
Rollout 46, Act: -1.071
U\)ĠwithĠrespectĠtoĠ\(x\)ĠandĠ\(y\)ĠyieldĠtheĠgravitationalĠandĠcentrifugalĠforces:ĊĠĠĠĠ
Rollout 75, Act: -1.076
ĠdifferentĠpositiveĠoddĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquaresĠofĠsixĠ(notĠnecessarilyĠĠdifferent)
Rollout 99, Act: -1.082
h_1Ġ&=Ġx_2Ġ-Ġx_1Ġ=Ġ0.5,Ġ\\Ċ\delta_
Rollout 80, Act: -1.126
ĠcoreĠbyĠ\(Ġ1^\circĠ\mathrm{C}Ġ\):**ĊĠĠĠ\[ĊĠĠĠ\text
Rollout 69, Act: -1.135
(a)sin(b)Ġ=Ġ[cos(aĠ-Ġb)Ġ-Ġcos(aĠ+Ġb)]/2.
Rollout 75, Act: -1.139
ĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquaresĠofĠsixĠ(notĠnecessarilyĠĠdifferent)ĠpositiveĠintegers.Ċ
Rollout 99, Act: -1.156
Ġ-Ġ\delta_0}{h_1Ġ+Ġh_0}Ġ\approxĠ-0.80
Rollout 50, Act: -1.197
ĠofĠbaseĠformula,ĠweĠexpressĠbothĠsidesĠinĠtermsĠofĠbaseĠ10:ĊĊ\[ĊyĠ=Ġ\
Rollout 1, Act: -1.254
ĠaĠHilbertĠspaceĠsuchĠthatĠtheĠimagesĠareĠorthonormal.ĊĊButĠhere,Ġwe'reĠdealingĠwithinĠthe
Rollout 44, Act: -1.275
.ĠForĠanĠidealĠgas,ĠinternalĠenergyĠdependsĠonlyĠonĠtemperature,ĠsoĠifĠnoĠworkĠisĠdoneĠandĠno
Rollout 75, Act: -1.304
ĠanyĠthreeĠpairwiseĠdifferentĠpositiveĠoddĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquaresĠofĠsixĠ(notĠnecessarily
Rollout 75, Act: -1.335
ĠthreeĠpairwiseĠdifferentĠpositiveĠoddĠintegersĠcanĠbeĠrepresentedĠasĠtheĠsumĠofĠtheĠsquaresĠofĠsixĠ(notĠnecessarilyĠ
Rollout 10, Act: -1.578
ĠimpartialĠgameĠunderĠtheĠnormalĠplayĠconditionĠisĠequivalentĠtoĠaĠnimber.ĠAlso,ĠallĠsubtractionĠgamesĠhaveĠperiodic

UP_PROJ

Top 16 Positive Activations
Rollout 95, Act: 2.665
1,...,2449ĠasĠperĠtheĠproblemĠstatement:ĊĊpointsĠAĠ=ĠP0;ĠP1
Rollout 94, Act: 2.650
ĠisĠNOTĠentangled.ĠSoĠanswerĠchoicesĠ(A)ĠandĠ(D)ĠincludeĠb,ĠbutĠ(D
Rollout 54, Act: 2.646
5,2,1,3Ġand4.ĠSoĠarrangedĠasĠ5,2,1,3
Rollout 54, Act: 2.643
Ġ4Ġ(methanol)ĊĊThisĠwouldĠmatchĠtheĠoptionsĠ(AĠ=Ġ5,Ġ2,Ġ
Rollout 54, Act: 2.639
)ĠâĨĴĠAnswerĠchoiceĠAĠorĠC.ĠTheĠoptionsĠgivenĠinĠtheĠproblem'sĠfirstĠpartĠ(A'sĠorder
Rollout 94, Act: 2.616
,ĠdĠisĠnotĠentangled.ĠSoĠ(A)ĠisĠwrong.ĊĊ(B)Ġa,cĠâĨĴĠbothĠa
Rollout 94, Act: 2.612
ĠisĠaĠcandidate.ĊĊ(C)Ġc,dĠâĨĴĠcĠisĠentangled,ĠdĠisĠnot.ĠSoĠ(C
Rollout 54, Act: 2.591
ĠfirstĠpartĠ(A'sĠorder):ĊĊAnswerĠoptionĠAĠsaysĠAĠ=5,2,1,3Ġand
Rollout 94, Act: 2.566
ĠNOTĠentangled.ĠSoĠanswerĠchoicesĠ(A)ĠandĠ(D)ĠincludeĠb,ĠbutĠ(D)
Rollout 94, Act: 2.565
.ĠSoĠanswerĠchoicesĠ(A)ĠandĠ(D)ĠincludeĠb,ĠbutĠ(D)ĠsaysĠa,b
Rollout 38, Act: 2.543
ĠthanĠlengthĠL,ĠwhichĠcontradictsĠtheĠproblem'sĠstatementĠthatĠtheĠlongerĠsideĠrunsĠdueĠwestĠ(i.e
Rollout 94, Act: 2.528
ĠSoĠanswerĠchoicesĠ(A)ĠandĠ(D)ĠincludeĠb,ĠbutĠ(D)ĠsaysĠa,b.
Rollout 94, Act: 2.515
Ġ(b)ĠisĠNOTĠentangled.ĠSoĠanswerĠchoicesĠ(A)ĠandĠ(D)ĠincludeĠb,
Rollout 76, Act: 2.511
1ĠandĠS2ĠareĠcolumnsĠ1-3,ĠbutĠS1ĠisĠcolumnsĠ1-2Ġand
Rollout 23, Act: 2.511
15ĠBq,ĠwhichĠisĠoptionĠ(D)Ġ117.555,Ġclose
Rollout 98, Act: 2.508
.ĊĊThus,Ġ3ĠvalidĠsystems.ĠButĠanswerĠoptionsĠareĠ(A)1,Ġ(B)3
Top 16 Negative Activations
Rollout 93, Act: -2.051
ĠcombinatorialĠexpressions.ĊĊLetĠmeĠverify:ĠConsiderĠtheĠsumĠSĠ=ĠΣ_{k=1}
Rollout 93, Act: -2.068
ĠwhenĠdealingĠwithĠsumsĠoverĠrootsĠofĠunityĠorĠtrigonometricĠsums,ĠsometimesĠsymmetryĠorĠpropertiesĠofĠcomplexĠnumbers
Rollout 93, Act: -2.069
,ĠexponentsĠwithĠcosineĠcanĠsometimesĠbeĠdealtĠwithĠusingĠpower-reductionĠformulas.ĠSinceĠtheĠexponentĠhereĠis
Rollout 78, Act: -2.079
.ĠLĠ/Ġ(v_wĠ+Ġ20)Ġ=Ġ1/60ĊĊTherefore,Ġfrom
Rollout 50, Act: -2.080
ĠlogarithmicĠidentitiesĠthatĠmightĠhelpĠhere.ĠTheĠchangeĠofĠbaseĠformulaĠisĠusefulĠforĠcomparingĠlogarithmsĠwithĠdifferent
Rollout 1, Act: -2.080
2).ĊĊSo,ĠhowĠdoĠweĠshowĠthatĠsuchĠaĠyĠexists?ĊĊInĠfiniteĠdimensions,ĠifĠIĠhave
Rollout 1, Act: -2.086
Ġy.ĠSoĠtheĠmainĠideaĠisĠtoĠfindĠaĠpointĠyĠsuchĠthatĠallĠxĠinĠSĠareĠatĠdistance
Rollout 81, Act: -2.110
Ġgeometry,ĠweĠcanĠmodelĠthis.ĊĊAlternatively,ĠmaybeĠcoordinateĠgeometryĠcanĠhelpĠhere.ĠLetĠmeĠplaceĠpointĠA
Rollout 91, Act: -2.131
.ĊĊButĠperhapsĠit'sĠbetterĠtoĠassignĠcoordinates.ĠLetĠmeĠplaceĠtheĠtriangleĠinĠcoordinateĠsystem.ĠLetĠme
Rollout 38, Act: -2.149
,ĠinitiallyĠheadingĠsouth,ĠandĠCaoĠbikesĠinĠaĠstraightĠlineĠacrossĠtheĠfieldĠtoĠaĠpointĠDĠon
Rollout 91, Act: -2.163
,ĠifĠtheĠmedianĠADĠmakesĠanĠangleĠofĠ45ĠdegreesĠwithĠsideĠBC,ĠthenĠfindĠtheĠvalue
Rollout 58, Act: -2.166
)Ġ=Ġ2018Ġ*ĠP(0).ĠWeĠneedĠtoĠsolveĠforĠk.ĊĊToĠcalculate
Rollout 50, Act: -2.181
ĠcomplicatedĠbecauseĠofĠallĠtheseĠlogarithmicĠterms.ĠMaybeĠIĠcanĠsimplifyĠeachĠlogarithmĠexpression?ĠLet'sĠexpand
Rollout 49, Act: -2.212
Ġstatement.ĊĊAlright,ĠsoĠgivenĠXĠisĠHausdorffĠandĠuncountable,ĠIĠneedĠtoĠfind
Rollout 72, Act: -2.239
ĠMaybeĠit'sĠbetterĠtoĠconvertĠthisĠintoĠequations.ĊĊLetâĢĻsĠrepresentĠÎ5ĠasĠ\(Ġe^{2
Rollout 93, Act: -2.275
,ĠperhapsĠweĠcanĠconsiderĠusingĠtheĠbetaĠfunctionĠorĠintegralĠapproximation.ĠButĠgivenĠthatĠtheĠexponentsĠareĠdiscrete

DOWN_PROJ

Top 16 Positive Activations
Rollout 95, Act: 2.588
P_{k-1}$ĠandĠ$P_{k+1}$ĠforĠ$kĠ=Ġ1,
Rollout 1, Act: 2.529
1Ġ+Ġr2Ġ>=ĠlĠandĠ|r1Ġ-Ġr2|Ġ<=Ġl.ĊĊInĠourĠcase,
Rollout 53, Act: 2.509
Ġproceed.ĠShoelaceĠformula:ĊĊSumĠoverĠiĠ(x_iĠy_{i+1}Ġ-Ġx
Rollout 14, Act: 2.321
ĠupĠaĠfunctional:ĊĊ$$Ċ\mathcal{L}Ġ=Ġ\left(Ġ\intĠf(x)
Rollout 14, Act: 2.248
Ġ\intĠf(x)ĠdxĠ\right)^3Ġ-Ġ\lambdaĠ\left(Ġ\intĠf(x)^
Rollout 53, Act: 2.209
.ĠShoelaceĠformula:ĊĊSumĠoverĠiĠ(x_iĠy_{i+1}Ġ-Ġx_{
Rollout 81, Act: 2.194
ĠalphaĠ+ĠtanĠbeta]/[1Ġ-ĠtanĠalphaĠtanĠbeta]ĊĊWeĠareĠgivenĠtan(angleĠBAC)=
Rollout 14, Act: 2.190
ĠfunctionalĠtoĠmaximizeĠisĠâĪ«fĠdxĠ-λĠâĪ«f2ĠdxĠ-ĠμĠâĪ«xĠf
Rollout 53, Act: 2.171
ĠShoelaceĠformula:ĊĊSumĠoverĠiĠ(x_iĠy_{i+1}Ġ-Ġx_{i
Rollout 16, Act: 2.164
inftyĠ(1/2)^jĠ=2,ĠwhichĠmatches.ĊĊSimilarly,ĠforĠn=2:
Rollout 89, Act: 2.161
SoĠfromĠhere,ĠmultiplyingĠbothĠsidesĠbyĠ5/3ĠtoĠisolateĠT:ĊĊTĠ=Ġ10
Rollout 34, Act: 2.156
ĠtoĠt.ĊĊd/dtĠ[x(t)]Ġ=Ġd/dtĠ[1]Ġ+Ġd/dt
Rollout 37, Act: 2.149
2-1)(3-1)(3-2)Ġ=Ġ1*2*1=2
Rollout 45, Act: 2.147
4Ġ+ĠbĠ-5Ġ=Ġ-9Ġ+Ġb,ĠwhichĠmatchesĠtheĠpreviousĠexpression.ĠSoĠbothĠagreements,
Rollout 14, Act: 2.126
,ĠweĠcanĠsetĠupĠaĠfunctional:ĊĊ$$Ċ\mathcal{L}Ġ=Ġ\left(Ġ\
Rollout 63, Act: 2.114
4}f(2,Ġ1)Ġ\)ĠbothĠsides:ĊĊ\[Ċ\frac{3}{
Top 16 Negative Activations
Rollout 79, Act: -2.063
Ġsee.ĠIfĠ"noticed"ĠisĠtheĠanagramĠindicator,ĠthenĠ"minorĠchanges"ĠhasĠtoĠbeĠan
Rollout 48, Act: -2.071
19E.ĊĊNow,ĠsumĠisĠ0xE19E.ĠNow,ĠaddĠtheĠlastĠword
Rollout 23, Act: -2.091
-32.ĠThenĠmultiplyĠbyĠ5e34:ĊĊN(t)=5e34Ġ*
Rollout 95, Act: -2.094
Ġ(-2,âĪļ671)/Ġ(5âĪļ3Ġ).ĊĊĠTheĠunitĠvectorĠinĠBCĠdirection:Ġ(-
Rollout 22, Act: -2.099
Ġ1e6ĠtoĠeliminateĠdecimals:ĊĊ18,168.7768Ġ*Ġ
Rollout 48, Act: -2.105
=57,758ĠâĨĴĠ0xE19EĠ(sinceĠ57,3
Rollout 14, Act: -2.119
^{-1/3}|ĠdxĠâī¤Ġ||f||_3Ġ||x^{1/3}||_
Rollout 80, Act: -2.121
Ġ*Ġ7.238ĠisĠ12.8Ġ*Ġ7Ġ=Ġ89.
Rollout 14, Act: -2.126
Ġ||f(x)x^{1/2}Ġ||_2Ġ*Ġ||x^{-1/2}||
Rollout 14, Act: -2.137
ThenĠHölderĠgives:ĊĊIĠâī¤Ġ||f||_2Ġ||1||_4Ġ||1||_
Rollout 4, Act: -2.162
ĠVĠseparately.ĊĊStartingĠwithĠC1:Ġ2000Ġ±Ġ10ĠpF.ĠThe
Rollout 14, Act: -2.199
Ġ\cdotĠ1Ġ\leĠ\|f\|_aĠ\|1\|_b,ĊĊButĠas
Rollout 14, Act: -2.225
Ġ||f(x)x^(-1/3)||_aĠ*||x^(1/3)||_
Rollout 48, Act: -2.269
612ĊĊSum:ĊĊStartingĠfromĠtheĠright:ĊĊ0Ġ+Ġ2Ġ=Ġ2ĊĊ0Ġ+
Rollout 46, Act: -2.364
ĠthatĠM1ĠisĠlocatedĠatĠ(Ġ-μ,Ġ0Ġ)ĠandĠM2ĠisĠatĠ(Ġ1
Rollout 70, Act: -2.825
ĠofĠNĠinĠbase-7Ġ(whichĠareĠ<=6),ĠsoĠtheĠleadingĠdigitĠofĠNĠ(base-

Layer 41

GATE_PROJ

Top 16 Positive Activations
Rollout 22, Act: 1.558
298ĠK)ĠgivesĠaboutĠ1,769Ġm/s.ĊĊGivenĠtheĠproblemĠsaysĠ"
Rollout 27, Act: 1.503
,ĠneedĠanotherĠvariable.ĊĊAlternatively,ĠcriticalĠtemperatureĠmayĠbeĠcalculatedĠviaĠequatingĠthermalĠenergyĠkĠT_cĠto
Rollout 27, Act: 1.497
Wait.ĠAlternativelyĠmaybe,ĠbecauseĠtheĠgivenĠÎĶE_rmsĠisĠ5.54e-1
Rollout 22, Act: 1.427
8ĠKĠ(25°C):âīĪ1,769Ġm/sĊĊAtĠ300
Rollout 46, Act: 1.343
ĠbeĠanĠobstacle.ĊĊAlternatively,ĠtheĠproblemĠmayĠbeĠconsideredĠinĠanĠinertialĠframeĠwhereĠUĠdoesĠnotĠinclude
Rollout 27, Act: 1.342
ĠtheĠkineticĠenergyĠperĠparticleĠtoĠcomputeĠtemperatureĠandĠlabelingĠthatĠasĠT_c.ĠSoĠproceedingĠunderĠthisĠassumption,
Rollout 46, Act: 1.317
,ĠthatĠwouldĠnotĠbeĠtheĠcase.ĊĊAlternatively,ĠmaybeĠtheyĠexpectĠrecognitionĠthatĠinĠtheĠco-rotating
Rollout 46, Act: 1.305
ĠhaveĠanĠerror,ĠandĠactuallyĠweĠareĠtoĠassumeĠthatĠtheĠtermsĠareĠcorrectĠasĠgradientsĠwithoutĠCoriolis
Rollout 46, Act: 1.288
ĠadditionalĠrelationships,ĠperhapsĠsimplifyingĠtheĠscenario.ĊĊAlternatively,ĠinĠsomeĠformulations,ĠmaybeĠUĠisĠtakenĠtoĠinclude
Rollout 3, Act: 1.286
8ĠKĠ(25°C)ĠyieldsĠ~168.5Ġm/s.ĠTherefore,ĠsinceĠthe
Rollout 3, Act: 1.264
ĠandĠroundedĠtoĠthreeĠsigĠfigsĠisĠ169Ġm/s.ĠThus,ĠpresentĠanswerĠboxedĠas
Rollout 3, Act: 1.245
compute",ĠsoĠit'sĠexpectedĠtoĠcompute.ĊĊAlternatively,ĠperhapsĠcheckĠifĠthereĠisĠanĠestablishedĠaverageĠspeedĠfor
Rollout 97, Act: 1.242
Ġt=ln(10)/k=4605.ĊĊThisĠisĠbecauseĠinĠtheĠfirstĠcase
Rollout 89, Act: 1.237
ĠteacherĠmayĠsuspectĠanswerĠisĠ66ĠorĠ67,ĠdependsĠonĠtheirĠapproach.ĊĊButĠgivenĠthat
Rollout 27, Act: 1.237
.ĠThereforeĠperhapsĠtheĠproblemĠisĠoversimplified,ĠassumingĠthatĠwithĠV_BeĠcalculated,ĠT_cĠisĠwhen
Rollout 3, Act: 1.235
ĠprecisionĠneeded.ĊĊAlternatively,ĠmaybeĠtheĠoriginalĠquestionĠassumesĠaĠspecificĠtemperatureĠbutĠforgotĠtoĠmention.ĠIfĠtheĠuser
Top 16 Negative Activations
Rollout 20, Act: -3.629
81=1,948,ĠĠ1800*1000=
Rollout 96, Act: -3.633
9*15=2535.Ġ2630Ġ-2535=
Rollout 60, Act: -3.657
ĠbyĠ328:ĊĊ328ĠgoesĠintoĠ332Ġonce,ĠremainderĠ4.
Rollout 90, Act: -3.673
0=201,800.Ġ224,000Ġ-20
Rollout 23, Act: -3.673
5Ġ*0.2=189.Ġ945*0.031âīĪ
Rollout 20, Act: -3.688
Ġ(46*20=920,Ġ920Ġ+46=96
Rollout 19, Act: -3.692
Ġ3081Ġmod83.ĊĊComputeĠ83*37=3071
Rollout 60, Act: -3.708
Ġ492ĊĊ492Ġ*Ġ0.01Ġ=Ġ4.92ĊĊ
Rollout 17, Act: -3.727
138.190,ĠandĠ0.1077*3.14
Rollout 95, Act: -3.768
3).ĊĊComputeĠarea:Ġ(1/2)|Ġ(0*(0Ġ-ĠâĪļ671/
Rollout 17, Act: -3.769
âīĪ138.190,ĠandĠ0.1077*3.1
Rollout 96, Act: -3.782
*3=85,683;Ġ85,683Ġ<88,
Rollout 13, Act: -3.783
0.57Ġ*Ġ6000Ġ=Ġ?ĊĊWait,ĠmaybeĠbetterĠtoĠprocessĠ60
Rollout 39, Act: -3.907
788=7092,ĠbecauseĠ9*700=6300
Rollout 17, Act: -4.065
41:Ġ8*1Ġ=8,Ġ8*0.41=3.2
Rollout 13, Act: -4.092
3000ĠâĨĴĠ0.57Ġ*Ġ6000Ġ=Ġ?ĊĊWait,Ġmaybe

UP_PROJ

Top 16 Positive Activations
Rollout 65, Act: 2.480
Ġk*a,Ġk*1995,Ġk*cĠinĠanyĠorder.ĊĊSo,ĠsupposeĠafterĠcutting
Rollout 68, Act: 2.470
.ĠAndĠsinceĠtheirĠproductĠpositive,ĠbothĠrootsĠareĠpositive.ĊĊTherefore,ĠinĠk>8,ĠquadraticĠequation
Rollout 0, Act: 2.438
ĠbutĠtheirĠexponentsĠareĠ1,ĠwhichĠisĠodd.ĠActually,ĠwaitĠnvm,ĠtheĠexponents
Rollout 62, Act: 2.417
?ĠBecauseĠ11*Ġp*qĠwouldĠbeĠthreeĠprimes,ĠbutĠnotĠfour.ĠUnlessĠfactorĠwithĠexponent
Rollout 39, Act: 2.403
Ġ990/9Ġ=Ġ110,ĠwhichĠisĠ11ĠfollowedĠbyĠaĠzero
Rollout 96, Act: 2.400
ĠdirectionĠofĠlineĠLĠisĠ(12/13,5/13).ĠTheĠunitĠvectorĠperpendicular
Rollout 21, Act: 2.378
ĠxĠ-f(x)=Ġ-0.5Ġ-0=-0.5.ĠCeil(-0.
Rollout 20, Act: 2.364
})Ġâī¡Ġ1*1*...1Ġâī¡Ġ1ĠmodĠ2021.
Rollout 9, Act: 2.356
).ĠTheĠaltitudeĠisĠh,ĠtheĠhorizontalĠdistanceĠfromĠGĠtoĠEM.ĠSoĠhĠisĠtheĠlengthĠweĠneed
Rollout 49, Act: 2.341
orĠsetĠisĠcompact,ĠsoĠit'sĠÏĥ-compact.ĠTherefore,ĠbeingĠaĠperfectĠsetĠinĠourĠX
Rollout 56, Act: 2.331
1ĠandĠâĪļ3,ĠhypotenuseĠ2,ĠanglesĠ90°,Ġ60
Rollout 57, Act: 2.322
ĠareĠalsoĠopen,ĠhenceĠallĠopenĠsetsĠareĠalsoĠclosed.ĠThus,ĠyouĠwouldĠhaveĠthatĠtheĠtopologyĠis
Rollout 62, Act: 2.317
:Ġ11*p*q*r,ĠwhichĠisĠfourĠprimes.ĠSoĠneedĠtheĠ(11k2
Rollout 65, Act: 2.316
ĠtheĠsmallerĠprismĠmustĠalsoĠbeĠkaĠ:ĠkbĠ:ĠkcĠforĠsomeĠkĠ<Ġ1.ĠTherefore,Ġthat
Rollout 49, Act: 2.312
ĠofĠaĠcompactĠmetrizableĠspaceĠisĠsigma-compact?ĠNO!ĠWait,Ġexample:ĊĊInĠtheĠCant
Rollout 56, Act: 2.308
-ĠHypotenuseĠA1A2Ġ=Ġ2.ĊĊSoĠcoordinates:ĊĊ-ĠLetâĢĻsĠsetĠA0
Top 16 Negative Activations
Rollout 57, Act: -2.566
Ġdon'tĠmanageĠtoĠfindĠtheĠseparationĠforĠregular,ĠperhapsĠIĠneedĠtoĠtryĠaĠdifferentĠapproach.ĠLetĠme
Rollout 24, Act: -2.594
ĠmuchĠconsiderationĠwithoutĠprogress,ĠIĠsuspectĠtheĠkeyĠinsightĠrequiresĠconstructingĠaĠinjectiveĠmappingĠbetweenĠrowsĠofĠtwoĠminimal
Rollout 10, Act: -2.599
Ġexist.ĠAsĠaĠcontestĠproblem,ĠtheĠsolutionĠmayĠbeĠfoundĠinĠrealizingĠtheĠperiodicityĠfailed,ĠbutĠthat
Rollout 14, Act: -2.600
ĠwhoĠsetĠtheĠproblemĠmustĠhaveĠanĠparticularĠmethodĠinĠmind.ĠLet'sĠgoĠback.ĊĊAnotherĠstrategy.ĠUse
Rollout 6, Act: -2.606
ĠinĠcirclesĠhere.ĠGivenĠtheĠtimeĠconstraints,ĠmaybeĠforĠproblemĠ3,ĠsinceĠit'sĠknownĠinĠcomplexity
Rollout 14, Act: -2.610
ĠtheĠpersonĠwhoĠsetĠtheĠproblemĠmustĠhaveĠanĠparticularĠmethodĠinĠmind.ĠLet'sĠgoĠback.ĊĊAnotherĠstrategy
Rollout 15, Act: -2.613
,ĠpowerĠconfigurationsĠmayĠmakeĠdifference.ĊĊAlternatively,ĠperhapsĠanswerĠisĠthatĠnĠmustĠĠbeĠaĠpowerĠof2
Rollout 15, Act: -2.632
ĠtheĠpossibleĠpatternsĠandĠtheirĠevolutions.ĠPerhapsĠaĠsolutionĠexistsĠwhereĠnãģĮĠisĠaĠpowerĠofĠtwo?
Rollout 14, Act: -2.634
ĠEitherĠway,ĠthisĠisn'tĠhelping.ĠPerhapsĠIĠneedĠaĠdifferentĠapproach.ĊĊAlternativeĠidea:ĠUseĠHölder
Rollout 37, Act: -2.639
Ġuncertain.ĠMoreover,ĠifĠgivenĠtheĠproblemĠĠmustĠanswerĠthenĠofficialĠsolutionĠusesĠsaidĠprimeĠconditions,ĠsoĠpossibly
Rollout 24, Act: -2.647
Ġproof,ĠandĠafterĠmuchĠconsiderationĠwithoutĠprogress,ĠIĠsuspectĠtheĠkeyĠinsightĠrequiresĠconstructingĠaĠinjectiveĠmappingĠbetween
Rollout 6, Act: -2.675
Ġtime.ĊĊSoĠperhapsĠtheĠanswerĠisĠnotĠstraightforward,ĠhenceĠwhyĠreferenceĠtoĠtheĠP/polyĠtheorem.ĠThe
Rollout 57, Act: -2.693
ĠmanageĠtoĠfindĠtheĠseparationĠforĠregular,ĠperhapsĠIĠneedĠtoĠtryĠaĠdifferentĠapproach.ĠLetĠmeĠrecallĠthat
Rollout 37, Act: -2.738
ĠkeyĠinsightsĠ(orĠlackĠthereof),ĠIĠthinkĠtheĠanswerĠhingesĠonĠtheĠfactĠthatĠdeterminant(A)Ġvanishes
Rollout 24, Act: -2.857
ĠforĠaĠproof,ĠandĠafterĠmuchĠconsiderationĠwithoutĠprogress,ĠIĠsuspectĠtheĠkeyĠinsightĠrequiresĠconstructingĠaĠinjective
Rollout 10, Act: -2.946
ĠinfinitelyĠmanyĠpositionsĠexist.ĠAsĠaĠcontestĠproblem,ĠtheĠsolutionĠmayĠbeĠfoundĠinĠrealizingĠtheĠperiodicityĠfailed

DOWN_PROJ

Top 16 Positive Activations
Rollout 48, Act: 2.825
=Ġ9+14=23.Ġ23ĠinĠhexĠisĠ17,Ġwhich
Rollout 39, Act: 2.818
092,ĠbecauseĠ9*700=6300,ĊĊ9*80
Rollout 74, Act: 2.642
ftĠdirectionĠisĠatĠθ,ĠandĠtheĠathwartshipsĠisĠatĠθĠ+Ġ90Ġdegrees
Rollout 96, Act: 2.606
3=3,211ĊĊSoĠ3.11...ĠHmm.ĠAnyway,ĠuseĠfractions
Rollout 32, Act: 2.495
Ġx3Ġ=Ġ1ĠareĠ1,ĠÏī,ĠandĠÏī2,ĠwhereĠÏīĠand
Rollout 34, Act: 2.487
)Ġ=Ġ-4Ġexp(0)Ġ+Ġexp(-t)Ġ+ĠCĠexp(t)ĊĊSo,ĊĊy
Rollout 96, Act: 2.417
=10,894.ĊĊ13*800=10,400
Rollout 96, Act: 2.401
ĠremainingĠ2561.Ġ13*197=2561.ĠYes
Rollout 22, Act: 2.387
458.933ĊĊNow,Ġ2,111Ġ*Ġ69Ġâī
Rollout 70, Act: 2.376
0.ĠThen,Ġ20ĠdividedĠby7=2,Ġremainder6.ĠThen,ĠdigitsĠare1
Rollout 25, Act: 2.360
ĠpreviousĠcaseĠwith:ĊĊOriginalĠmuĠ=ĠGM_earth,ĠleadingĠtoĠperigeeĠ9898
Rollout 70, Act: 2.359
7=1Ġrem2,Ġ2/7=0Ġrem2.ĠSoĠdigitsĠ3Ġ1
Rollout 36, Act: 2.354
^4Ġ-30,624s2Ġ+140,541,6
Rollout 16, Act: 2.346
ĠisĠconvertedĠonĠorĠbeforeĠdayĠtĠisĠ1Ġ-Ġ(1/2)^{t+1}
Rollout 76, Act: 2.335
-95:Ġ512Ġ-90=422,Ġ422Ġ-
Rollout 11, Act: 2.318
1,Ġ1/aĠ=Ġbc,Ġ1/bĠ=Ġac,Ġ1/cĠ=Ġab,Ġas
Top 16 Negative Activations
Rollout 10, Act: -1.951
ĠbetweenĠ3ĠandĠ2?ĠLet'sĠsee:ĊĊ0Ġto2:Ġ+2ĊĊ2Ġto5
Rollout 81, Act: -1.954
h/-3=Ġh/3.ĠSoĠsinceĠrunĠisĠnegativeĠdirection,ĠfromĠAĠtoĠBĠdeltaĠx
Rollout 77, Act: -1.965
Ġ*Ġf(14,10)=Ġ(24/10)*70=24
Rollout 77, Act: -1.976
8.ĠWhichĠisĠLCM(14,24)=168.ĠIndeed.ĠThenĠf
Rollout 73, Act: -1.980
ĠtotalĠsumĠisĠ2036.ĠCorrect.ĊĊSoĠ18434-20
Rollout 27, Act: -1.990
ĠgivenĠby:ĊĊT_cĠ=Ġ(Ġ(n^(2/3)Ġh^2)Ġ/Ġ(2
Rollout 91, Act: -1.995
)Ġ=Ġ[2(tĠ+1)]Ġ/Ġ[2Ġsqrt(Ġ(tĠ+1)^2Ġ+Ġt
Rollout 71, Act: -2.020
ĠyieldĠanswers,ĠthisĠtripletĠisĠlikelyĠtheĠonlyĠpossibleĠone.ĠHence,ĠabsoluteĠsumĠ|39|Ġ+
Rollout 78, Act: -2.035
ĠgoingĠeastĠatĠ20Ġmph,ĠSteveĠisĠgoingĠwestĠatĠ20Ġmph.ĠThereĠareĠtwo
Rollout 82, Act: -2.049
)/69=Ġ(16+30=46;46+36=8
Rollout 56, Act: -2.101
ļ3).ĊĊDistanceĠA4A2:ĊĊxĠdifference:Ġ0Ġ-7/6=Ġ-7
Rollout 4, Act: -2.159
Ġ1200.ĠYep,Ġthat'sĠright.ĠSo,ĠC_eqĠisĠ120
Rollout 33, Act: -2.195
120=120+48=168,Ġ168+3
Rollout 81, Act: -2.287
0^2Ġ=Ġ4900ĊĊ4*11*561Ġ=Ġ4
Rollout 73, Act: -2.355
0:ĊĊ10Ġ+18=28ĊĊ28Ġ+32=60ĊĊ6
Rollout 73, Act: -2.512
8ĊĊ28Ġ+32=60ĊĊ60Ġ+56=116ĊĊ

Layer 42

GATE_PROJ

Top 16 Positive Activations
Rollout 37, Act: 1.339
ĠĠpositiveĠintegersĠ(positiveĠintegerĠexcludesĠĠ$0$Ġ).ĠLetĠĠ$$ĠA=\begin{pmatrix
Rollout 86, Act: 1.313
.ĠThatĠmeansĠweĠcanĠuseĠeachĠfunctionĠmoreĠthanĠonce,Ġright?ĠTheĠgoalĠisĠtoĠchooseĠfourĠfunctions
Rollout 84, Act: 1.295
Ġit,ĠandĠmĠandĠnĠareĠcoprimeĠpositiveĠintegers.ĠThen,ĠifĠheĠhadĠpurchasedĠhalfĠas
Rollout 51, Act: 1.282
ĠbeĠaĠconstantĠnumberĠsuchĠthatĠĠ$0\leqĠ\thetaĠ\leqĠ\pi.$ĠčĊ
Rollout 50, Act: 1.275
[\log_{20x}Ġ(22x)=\log_{2x}Ġ(20
Rollout 72, Act: 1.258
ĠnĠ\)ĠforĠwhichĠthereĠexistĠthreeĠ(notĠnecessarilyĠdistinct)ĠrootsĠofĠunityĠofĠorderĠ\(ĠnĠ\
Rollout 86, Act: 1.246
Ġallowed.ĠThatĠmeansĠweĠcanĠuseĠeachĠfunctionĠmoreĠthanĠonce,Ġright?ĠTheĠgoalĠisĠtoĠchooseĠfour
Rollout 13, Act: 1.207
Ġ4Ġyears.ĠHowĠmuchĠmoreĠmoneyĠwillĠsheĠoweĠifĠtheĠinterestĠcompoundsĠquarterlyĠthanĠifĠtheĠinterestĠcompounds
Rollout 86, Act: 1.200
,\Ġ\sinĠx,\Ġ\cosĠx$ĠĠrepeatedly.ĠFindĠtheĠminimumĠvalueĠofĠĠ$\int_{
Rollout 80, Act: 1.191
mathrm{~g}Ġ/Ġ\mathrm{cm}^{3}$,ĠandĠaĠspecificĠheatĠofĠ$
Rollout 6, Act: 1.175
{N}Ġ\mapstoĠ\mathbb{N}$ĠĠsuchĠthatĠforĠeachĠĠ$nĠ\
Rollout 65, Act: 1.168
P_{},$ĠandĠbothĠofĠwhichĠhaveĠnonzeroĠvolume.ĠĠGivenĠthatĠ$b=199
Rollout 28, Act: 1.151
Ġterm:ĊĊFirstĠterm:Ġ(u2Ġ+Ġ2uvĠ+v2)/4ĊĊSecondĠterm:
Rollout 85, Act: 1.123
)ĠmustĠformĠtheĠsetĠ{n,Ġn+1,Ġn+2}ĠforĠsomeĠnonnegativeĠinteger
Rollout 86, Act: 1.116
ĠrepetitionĠallowed.ĠThatĠmeansĠweĠcanĠuseĠeachĠfunctionĠmoreĠthanĠonce,Ġright?ĠTheĠgoalĠisĠtoĠchoose
Rollout 86, Act: 1.093
Ġ{x,ĠsinĠx,ĠcosĠx},ĠwithĠrepetitionĠallowed.ĠThatĠmeansĠweĠcanĠuseĠeachĠfunctionĠmore
Top 16 Negative Activations
Rollout 4, Act: -2.774
9ĠJ=15ĠnJ.ĊĊSimilarly,ĠcomputeĠE_max=0.5*120
Rollout 28, Act: -2.775
,Ġk)Ġsatisfying:ĊĊFromĠtheĠcenterĠequations:ĊĊ1.Ġ2AĠhĠ+ĠBĠkĠ+ĠD
Rollout 8, Act: -2.775
0P5).ĊĊLetĠmeĠtryĠthisĠapproach.ĊĊFirst,ĠhowĠmanyĠcolorĠsequencesĠareĠ"happy"?ĊĊCase
Rollout 96, Act: -2.782
Ġβ.ĊĊSoĠperhapsĠsolvingĠtheĠproblemĠviaĠprojection.ĊĊFirst,ĠprojectionĠofĠPĠontoĠlineĠL.ĊĊLineĠL
Rollout 8, Act: -2.792
),ĠorĠallĠredsĠorĠallĠgreens.ĊĊSoĠfirst,ĠweĠcanĠcalculateĠtheĠnumberĠofĠsuchĠhappyĠsequences
Rollout 97, Act: -2.796
ĠLetâĢĻsĠletĠtheĠamountĠdissociatedĠatĠtimeĠtĠbeĠsuchĠthatĠtheĠpartialĠpressureĠofĠN2O5
Rollout 48, Act: -2.814
ĠDoesĠtheĠheaderĠcontainsĠerror?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠIPv4Ġheader
Rollout 48, Act: -2.822
ĠtheĠone'sĠcomplement.ĊĊAlternatively,Ġhere'sĠaĠstep-by-stepĠprocess:ĊĊStartĠwithĠaĠ32
Rollout 33, Act: -2.827
Ġthere'sĠnoĠk=1...0Ġcondition.ĊĊForn=2:ĊĊPermutations[1,2
Rollout 97, Act: -2.868
Ġ1Ġatm.ĠAtĠtimeĠt,ĠletĠtheĠextentĠofĠreactionĠbeĠxĠ(inĠtermsĠofĠpressure).
Rollout 56, Act: -2.869
:Ġ(0,Ġ-âĪļ3/2)ĊĊToĠcomputeĠtheĠareaĠofĠtriangleĠA1A2A
Rollout 86, Act: -2.887
.ĊĊSimilarly,ĠintegralĠofĠx2Ġcos2xĠdxĠcanĠbeĠhandledĠanalogously.ĊĊUsingĠcos2x
Rollout 73, Act: -2.908
}(10Ġ-i)2^i]ĊĊFirstĠterm:Ġsum_{i=0}^{1
Rollout 93, Act: -2.943
ĠweĠtryĠtoĠseeĠaĠpatternĠhere.ĊĊmĠsum:ĊĊ1:1,Ġ1=2^0
Rollout 48, Act: -3.026
0Ċ7.Ġ0A00Ġ=Ġ0x0A00Ċ8.Ġ
Rollout 20, Act: -3.169
47.ĊĊCalculatingĠthat:ĊĊMultiplyĠstepĠbyĠstep:ĊĊ2*3=6ĊĊ6*7

UP_PROJ

Top 16 Positive Activations
Rollout 43, Act: 2.958
ĠCĠlistsĠexactlyĠthis.ĠSoĠIĠthinkĠtheĠanswerĠisĠoptionĠC.ĊĊ**FinalĠAnswer**Ċ\boxed{
Rollout 8, Act: 2.939
Ġ31/126,ĠsoĠm=31,Ġn=126.ĊĊWait
Rollout 84, Act: 2.900
ĠheĠpurchased.ĠSoĠtheĠanswerĠmustĠbeĠ37.ĊĊTherefore,ĠconfidentĠthatĠtheĠanswerĠisĠ\boxed
Rollout 74, Act: 2.842
ĠdeviationĠisĠsmall,ĠthenĠtheĠapproximationĠδθĠâīĪĠ(H_bĠsinθĠ+ĠH_sĠcosθ
Rollout 92, Act: 2.811
,ĠtheĠanswerĠshouldĠbeĠapproximatelyĠ+3.78Ġdiopters,ĠroundedĠtoĠtwoĠdecimalĠplaces,
Rollout 84, Act: 2.743
ĠofĠit,ĠandĠmĠandĠnĠareĠcoprimeĠpositiveĠintegers.ĠThen,ĠifĠheĠhadĠpurchasedĠhalf
Rollout 73, Act: 2.741
ĠtheĠremainderĠisĠ398.ĠSoĠanswerĠisĠ398.ĊĊWait,ĠbutĠjustĠto
Rollout 19, Act: 2.738
Ġintegersn.ĠSoĠseemsĠtheĠanswerĠisĠjust78.ĠButĠwait,ĠletĠmeĠcheckĠforĠsmallerĠnumbers
Rollout 84, Act: 2.731
Ġ37.ĊĊTherefore,ĠconfidentĠthatĠtheĠanswerĠisĠ\boxed{37}.ĊĊ**FinalĠAnswer
Rollout 40, Act: 2.720
1421356ĠisĠ19.ĠTherefore,ĠtheĠanswerĠisĠ19.ĊĊ
Rollout 5, Act: 2.710
ĠtheĠgreatestĠintegerĠnotĠexceedingĠthatĠisĠ181.ĠTherefore,ĠtheĠanswerĠisĠ181
Rollout 8, Act: 2.696
126,ĠsoĠm=31,Ġn=126.ĊĊWait,ĠbutĠlet's
Rollout 5, Act: 2.694
181.25ĠisĠ181.ĠTherefore,ĠfinalĠanswerĠisĠ181
Rollout 92, Act: 2.691
ĠroundingĠtoĠtwoĠdecimalĠplaces,Ġ3.78ĠD.ĊĊAlternatively,ĠtheĠproblemĠmightĠbeĠconsideringĠthe
Rollout 5, Act: 2.689
ĠtheĠgreatestĠintegerĠnotĠexceedingĠthisĠisĠ181.ĊĊWait,ĠthisĠseemsĠstraightforwardĠnow,ĠbutĠmaybe
Rollout 78, Act: 2.683
ĠasĠm/nĠwhereĠmĠandĠnĠareĠcoprime,ĠandĠthenĠcomputeĠmĠ+Ġn.ĊĊAlright,
Top 16 Negative Activations
Rollout 38, Act: -1.834
ĠSolvingĠtheĠquadraticĠequationĠafterĠsquaringĠbothĠsides.ĊĊ6.Ġ**FindingĠRatios**:ĊĠĠĠ-
Rollout 95, Act: -1.840
3,Ġ2/3).ĊĊĠWait,Ġwrong.ĠPerpendicularĠvectorĠtoĠ(-2/3,ĠâĪ
Rollout 93, Act: -1.845
theta\)ĠasĠ\(\frac{e^{i\theta}Ġ+Ġe^{-i\theta}}{
Rollout 13, Act: -1.846
ĠthousandthsĠplaceĠshiftedĠbyĠfourĠdecimalĠplacesĠforĠmultiplierĠposition):Ċ0.056275Ċ
Rollout 11, Act: -1.848
\)ĠandĠanalyzeĠtheĠsymmetryĠandĠpropertiesĠofĠtheseĠroots.ĊĊ3.Ġ**SymmetricĠandĠRealĠConditions**:Ċ
Rollout 14, Act: -1.861
AnotherĠidea:ĠForĠeachĠx,ĠuseĠtheĠAM-GMĠinequality.ĊĊSupposeĠforĠeachĠx,Ġwe
Rollout 65, Act: -1.879
ĠleadsĠtoĠequationsĠfromĠmaintainingĠsimilarityĠrequiresĠthat,ĠthroughĠsubstitutionĠofĠproportionality.ĊĊSo,ĠinĠcaseĠcuttingc
Rollout 11, Act: -1.901
ĠandĠcyclicĠinĠ\(a,Ġb,Ġc\).ĊĊ2.Ġ**TransformationĠandĠSimplification**:ĊĠĠ
Rollout 33, Act: -1.915
ĠfromĠtheĠtotalĠpermutationsĠandĠadjustĠusingĠinclusion-exclusion.ĊĊ2.Ġ**ApplyingĠInclusion-Exclusion
Rollout 75, Act: -1.916
ĠcanĠbeĠeitherĠ0,Ġ1,ĠorĠ4ĠmoduloĠ8.ĊĊ2.Ġ**ExampleĠDecom
Rollout 33, Act: -1.938
m=5)**:ĠIntersectionĠofĠallĠ\(A_1,ĠA_2,ĠA_3,
Rollout 75, Act: -1.940
)/4=0mod4?ĠNo,Ġno:ĊĊWait,ĠletâĢĻsĠcalculateĠ2k.ĊĊActually
Rollout 70, Act: -1.969
i=0}^kĠd_iĠ\cdotĠ7^iĊĠĠĠĠĠ\]ĊĠĠĠ-ĠThis
Rollout 11, Act: -1.993
Ġroots.ĊĊ3.Ġ**SymmetricĠandĠRealĠConditions**:ĊĠĠĠ-ĠSinceĠ\(p\)ĠandĠ\(q
Rollout 50, Act: -2.047
ĠtheĠratioĠofĠtheĠequationsĠderivedĠfromĠtheĠlogarithmicĠterms.ĠWeĠfindĠthat:ĊĊ\[Ċ22x
Rollout 85, Act: -2.134
ĠdetermineĠtheĠlabelsĠforĠadjacentĠpointsĠandĠobserveĠanyĠpatterns.ĊĊ2.Ġ**AnalyzingĠInitialĠPoints**:ĊĠĠ

DOWN_PROJ

Top 16 Positive Activations
Rollout 60, Act: 2.434
)ĊĊThus,ĠsubstitutingĠf'ĠintoĠf'':ĊĊf''Ġ=Ġf0Ġ*Ġ(vĠ+Ġv
Rollout 0, Act: 2.430
ĠofĠsuchĠcoprimeĠpairsĠisĠ\(2^k\),ĠwhereĠ\(k\)ĠisĠtheĠnumberĠof
Rollout 20, Act: 2.383
47ĠareĠcoprime,ĠtheĠChineseĠRemainderĠTheoremĠsaysĠthatĠtheĠcongruenceĠmodĠ
Rollout 54, Act: 2.358
ĠonĠO,ĠwhileĠpropionate'sĠchargeĠisĠdelocalized,ĠmakingĠitĠaĠweakerĠnucleophile.ĊĊSo
Rollout 91, Act: 2.325
ĠBĠandĠCĠusingĠtheĠdotĠproductĠformula.ĊĊDotĠproductĠformula:Ġcos(theta)Ġ=Ġ(vĠâĭ
Rollout 22, Act: 2.287
Ġ2,111Ġ*Ġ69ĠâīĪĠ145,ĠĠsoĠ6
Rollout 15, Act: 2.256
ĠmatrixĠisĠnilpotentĠifĠsomeĠpowerĠofĠitĠisĠtheĠzeroĠmatrix.ĊĊSo,ĠforĠMĠbeingĠthe
Rollout 49, Act: 2.237
ĠanyĠsubsetĠAĠofĠX,ĠwhenĠgivenĠtheĠsubspaceĠtopology,ĠisĠÏĥ-compact.ĊĊInĠparticular,
Rollout 20, Act: 2.236
Ġ47.ĠTherefore,ĠbyĠtheĠChineseĠRemainderĠTheorem,ĠsolvingĠÏĥ(an)Ġâī¡Ġ
Rollout 74, Act: 2.183
ĠtheĠshipĠisĠheadedĠatĠanĠangleĠθĠfromĠmagneticĠnorth,ĠthenĠtheĠship'sĠmagneticĠfieldsĠwouldĠbe
Rollout 82, Act: 2.170
ĠâīĪĠ0.2319ĊĊp2Ġ=Ġ15/69ĠâīĪ
Rollout 59, Act: 2.167
ĠandĠBĠheads,ĠCĠtails:Ġ(1/2)(1/2)(3/7)Ġ=
Rollout 69, Act: 2.160
rationals.ĠTherefore,ĠtheĠcoefficientsĠofĠtheĠirrationalĠtermsĠmustĠbeĠzero,ĠandĠtheĠrationalĠpartĠmustĠequal
Rollout 54, Act: 2.136
-stabilizedĠanion;ĠtheĠchargeĠisĠdelocalized,ĠwhichĠmakesĠitĠaĠweakerĠnucleophileĠcompared
Rollout 23, Act: 2.129
ĠbutĠifĠanswerĠoptionĠ(D)ĠisĠ117.555ĠBq,ĠperhapsĠit
Rollout 20, Act: 2.129
*3*7.ĊĊCheckĠif2^42âī¡1mod43Ġyes.ĊĊButĠcheckĠorder
Top 16 Negative Activations
Rollout 52, Act: -2.075
Ġ+1)/2=Ġ(38Ġ-105)/2=Ġ(-67)/2=
Rollout 31, Act: -2.077
ĠlinearĠcombinationĠofĠa_{0}ĠandĠa_{1}?ĊĊWaitĠhere'sĠanĠangleĠtoĠapproach:
Rollout 13, Act: -2.083
55ĊMultiplyĠasĠnumbers:ĊĊĠĠ11255Ġ(ignoreĠdecimalsĠforĠnow)Ċx
Rollout 85, Act: -2.094
ĠareĠ2000*2Ġ+2024Ġ=6024ĠandĠ2
Rollout 82, Act: -2.106
ĠcodeĠisĠjustĠtheĠinitialĠsymbol,ĠnoĠbinaryĠdigits.ĠSoĠlen_binaryĠ=Ġ0.ĠTherefore,Ġcontribution
Rollout 97, Act: -2.107
Ġ2ξ)Ġ+Ġ2ξĠ+ĠξĠ=Ġn0Ġ+Ġξ.ĠSoĠtotalĠpressure
Rollout 72, Act: -2.127
1Ġ+Ġ1Ġ-Ġ1Ġ=Ġ1Ċ-Ġ1Ġ-Ġ1Ġ-Ġ1Ġ=
Rollout 75, Act: -2.128
4+4+1+4+4+1=16+4+4+1+
Rollout 62, Act: -2.133
Ġ+22kĠ+1Ġ-11kĠ-1Ġ+11=121k2
Rollout 66, Act: -2.140
12n2Ġ+n,ĠthenĠmĠandĠnĠwouldĠhaveĠaĠparityĠdeterminedĠbyĠn.ĠIfĠn
Rollout 48, Act: -2.171
Ċ4.Ġ4000Ċ5.Ġ4006Ċ6.Ġ0
Rollout 34, Act: -2.200
Ġe^{-t}Ġ+3Ġe^tĠ+Ġe^{-t}Ġ-4Ġ=Ġ3Ġe^
Rollout 43, Act: -2.212
0⣩Ġ+Ġβ|1⣩Ġ-Ġγ|1⣩)ĠâĬĹĠ|0⣩,ĠwhichĠsimpl
Rollout 94, Act: -2.234
0>Ġ+Ġ|01>Ġ-Ġ|10>Ġ-Ġ|11>Ġ)ĊĊAgain,Ġignoring
Rollout 14, Act: -2.280
$.ĠSoĠ$f(x)Ġ=ĠaĠ-ĠbĠx$.ĠĊĊButĠwait,ĠweĠareĠassumingĠthatĠ$
Rollout 79, Act: -2.317
ĠH,ĠA,ĠN,ĠG,ĠE,ĠS.ĠSoĠSĠisĠthereĠonce.ĠOkay.ĠSo

Layer 43

GATE_PROJ

Top 16 Positive Activations
Rollout 22, Act: 2.570
{1694}Ġm/s.ĊĊ<|im_start|>answerĊToĠcomputeĠtheĠmeanĠmolecularĠspeedĠ\(ĠvĠ\
Rollout 50, Act: 2.525
ĠAnswer**Ċ\boxed{112}ĊĊ<|im_start|>answerĊGivenĠtheĠequation:ĊĊ\[Ċ\log_{
Rollout 59, Act: 2.510
ĠAnswer**Ċ\boxed{515}ĊĊ<|im_start|>answerĊJackieĠandĠPhilĠeachĠflipĠthreeĠcoins:
Rollout 9, Act: 2.507
FinalĠAnswer**Ċ\boxed{25}ĊĊ<|im_start|>answerĊSquareĠ\(AIME\)ĠhasĠsidesĠofĠlength
Rollout 0, Act: 2.503
ĠAnswer**Ċ\boxed{128}ĊĊ<|im_start|>answerĊGivenĠaĠrationalĠnumberĠbetweenĠ0ĠandĠ
Rollout 65, Act: 2.500
FinalĠAnswer**Ċ\boxed{40}ĊĊ<|im_start|>answerĊGivenĠaĠrightĠrectangularĠprismĠ\(ĠPĠ\)
Rollout 20, Act: 2.484
Ġ\(\boxed{125}\).ĊĊ<|im_start|>answerĊToĠsolveĠtheĠproblem,ĠweĠneedĠtoĠfind
Rollout 71, Act: 2.478
FinalĠAnswer**Ċ\boxed{98}ĊĊ<|im_start|>answerĊGivenĠtheĠpolynomialĠ\(x^3Ġ-Ġ
Rollout 77, Act: 2.473
ĠAnswer**Ċ\boxed{364}ĊĊ<|im_start|>answerĊGivenĠtheĠfunctionĠ\(ĠfĠ\)ĠdefinedĠon
Rollout 8, Act: 2.464
ĠAnswer**Ċ\boxed{157}ĊĊ<|im_start|>answerĊKathyĠhasĠ5ĠredĠcardsĠandĠ
Rollout 34, Act: 2.427
Ġe^{-t}Ġ-Ġ4}\).ĊĊ<|im_start|>answerĊToĠsolveĠtheĠgivenĠintegralĠequationsĠforĠcontinuousĠfunctions
Rollout 61, Act: 2.427
ĠAnswer**Ċ\boxed{913}ĊĊ<|im_start|>answerĊGivenĠtheĠsetĠ\(ĠSĠ\)ĠofĠintegers
Rollout 91, Act: 2.422
**FinalĠAnswer**Ċ\boxed{2}ĊĊ<|im_start|>answerĊGivenĠtriangleĠ\(ABC\)ĠwithĠmedianĠ\(AD
Rollout 70, Act: 2.413
ĠAnswer**Ċ\boxed{315}ĊĊ<|im_start|>answerĊAĠpositiveĠintegerĠ\(ĠNĠ\)ĠisĠcalled
Rollout 55, Act: 2.412
}{2}Ġ\end{pmatrix}}ĊĊ<|im_start|>answerĊTheĠgivenĠmatrixĠ\(\mathbf{A
Rollout 62, Act: 2.400
ĠAnswer**Ċ\boxed{132}ĊĊ<|im_start|>answerĊToĠfindĠtheĠleastĠpositiveĠintegerĠ\(ĠmĠ\
Top 16 Negative Activations
Rollout 61, Act: -3.123
3Ġ+7Ġ+1=ĠindeedĠlet'sĠcompute:ĊĊ37Ġ+Ġ31ĠisĠ68
Rollout 71, Act: -3.123
.Ġ892=7921,Ġ902=8100.
Rollout 56, Act: -3.130
StepĠ0:Ġ~0.866ĊĊStepĠ1:Ġ~1.299
Rollout 35, Act: -3.141
inant:Ġ49Ġ+4=53,Ġnon-integerĠroots.ĊĊD=-1:Ġb
Rollout 67, Act: -3.142
3=Ġ-192Ġ+540=348.ĠPerfect.ĊĊThusĠsoĠtheĠvalues
Rollout 71, Act: -3.146
.Ġ782=6084,Ġ792=6241.
Rollout 48, Act: -3.157
12ĠisĠ46,610.Ġ17,204Ġ+Ġ4
Rollout 56, Act: -3.180
StepĠ1:Ġ~1.299ĊĊStepĠ2:Ġ~1.511
Rollout 71, Act: -3.184
6,Ġ452=2025.Ġ1984/4=4
Rollout 71, Act: -3.185
(136)=11.66,ĠnotĠinteger.ĊĊWait,ĠperhapsĠlookĠforĠeĠsuch
Rollout 30, Act: -3.189
000Ġ+28,210=4,058,210,
Rollout 71, Act: -3.218
?Ġ372=1369,Ġ382=1444,
Rollout 72, Act: -3.239
Ġ1Ġ-0.5=0.5.ĠHmm.ĊĊAlternatively,ĠconsiderĠtheĠimaginaryĠparts.ĠWait
Rollout 71, Act: -3.280
,Ġ452=2025.ĠSoĠ2008ĠisĠnotĠaĠsquare
Rollout 11, Act: -3.298
Ġa*b*cĠ=k*k*(1/k2)Ġ=Ġ1.ĠAlso,ĠnoneĠofĠthemĠare
Rollout 71, Act: -3.348
.Ġ442=1936,Ġ452=2025.

UP_PROJ

Top 16 Positive Activations
Rollout 49, Act: 2.845
ĠtoĠrecallĠthatĠinĠaĠfirst-countable,ĠseparableĠspace,ĠeveryĠopenĠcoverĠhasĠaĠcountableĠsub
Rollout 62, Act: 2.657
)ĠplusĠtheĠfactorsĠinĠtheĠotherĠnumber.ĊĊWait,ĠifĠ11ĠisĠaĠprimeĠfactor,Ġthe
Rollout 33, Act: 2.646
ĠpositionĠasĠ6!Ġ=Ġ720,ĠbutĠactuallyĠtheĠfirstĠfiveĠelementsĠmustĠbeĠexactlyĠ
Rollout 56, Act: 2.642
useĠisĠoppositeĠtheĠrightĠangle.ĠWait,Ġno,ĠhypotenuseĠisĠtheĠsideĠoppositeĠtheĠrightĠangle
Rollout 82, Act: 2.631
âīĪ3.105ĠperĠcodeword?ĠWait,Ġno:Ġgroup3'sĠcodewords
Rollout 8, Act: 2.627
!Ġ)Ġ/Ġ(5Ġ-Ġ(5Ġ-r))!Ġ;Ġwait,ĠmaybeĠit'sĠbetterĠtoĠthink
Rollout 58, Act: 2.613
i/(iĠ+Ġ1))^{i}Ġ]...ĠWait,ĠmaybeĠnot.ĊĊWait,ĠtheĠentire
Rollout 21, Act: 2.548
Ġx,Ġf(x)Ġ-ĠnĠ=Ġ0ĠforĠsomeĠnĠâĪĪĠZ.ĠButĠagain,ĠequationsĠneed
Rollout 79, Act: 2.538
Ġletters:Ġ"Incident"ĠhasĠ8Ġletters,ĠbutĠ"Room"ĠisĠ4.ĠLet's
Rollout 43, Act: 2.536
ĠdidĠtheyĠmentionĠthatĠtheĠtargetĠisĠtheĠfirstĠqubitĠinsteadĠofĠtheĠsecond?ĠLetĠmeĠcheck.ĊĊThe
Rollout 82, Act: 2.531
Ġ*ĠtotalĠprobabilityĠofĠtheĠgroup).ĠWait,Ġno,ĠtheĠtotalĠexpectedĠnumberĠisĠsum_{i=1
Rollout 65, Act: 2.524
1,Ġc.,ĠneedĠproportionsĠeither:ĊĊĠa/kĠa=ĠB1Ġ/k*1995
Rollout 66, Act: 2.522
,Ġno,ĠallĠthreeĠareĠpairwiseĠcoprime?ĠWait,Ġno.ĠWait,ĠhypotenuseĠmust
Rollout 56, Act: 2.518
otenuseĠisĠoppositeĠtheĠrightĠangle.ĠWait,Ġno,ĠhypotenuseĠisĠtheĠsideĠoppositeĠtheĠright
Rollout 21, Act: 2.516
(x)Ġ-Ġfloor(x)Ġ=Ġc,ĠaĠconstant,ĠthenĠf(xĠ+1)Ġ-f(x
Rollout 56, Act: 2.491
ĠisĠtheĠsideĠoppositeĠtheĠrightĠangle.ĠWait,Ġno:ĠinĠaĠright-angledĠtriangle,ĠtheĠsides
Top 16 Negative Activations
Rollout 38, Act: -2.045
ĊĊThus,ĠEquation1.ĊĊEquation2Ġremains:ĊĊ(WĠ+x)/6.2=Ġsqrt(x
Rollout 81, Act: -2.053
21)^2ĠâĢĵ17,242*(2601/121)+
Rollout 45, Act: -2.062
Ġ3a]ĊĊEquation4:4kĠ+2mĠ+Ġ(4mĠ-16k
Rollout 67, Act: -2.063
ĠBnĠ+ĠC\).ĊĊUsingĠtheĠgivenĠterms:Ċ-ĠForĠ\(n=1\):Ġ\(1
Rollout 4, Act: -2.067
25.2004=603*25Ġ+603*0.
Rollout 67, Act: -2.070
*8Ġ+1020=Ġ-72*64Ġ+Ġ3936Ġ+
Rollout 7, Act: -2.076
5*(2N/3)Ġ=Ġ(75*2)/3Ġ*N=150
Rollout 45, Act: -2.101
EquationĠ3]ĊĊFromĠf(2):4kĠ+2mĠ+n=12Ġ[
Rollout 67, Act: -2.119
Ġ+Ġd)(pĠ+Ġq)Ġ=Ġ1716.Ġ...(2)ĊĊForĠn=3
Rollout 96, Act: -2.123
16*148=2316*(100+40+8)=
Rollout 34, Act: -2.125
)Ġ=Ġ\int_{0}^{t}Ġe^{-2(tĠ-Ġs)}Ġ[2x(s
Rollout 12, Act: -2.144
Ġ+Ġu)^2Ġ-Ġν2Ġ)Ġ(1Ġ-Ġ(νĠ+Ġu)^2Ġ)Ġ)ĊĊSimplify
Rollout 81, Act: -2.154
01*484=2601*400Ġ+2601*
Rollout 85, Act: -2.186
ĠisĠtoĠfigureĠoutĠallĠpossibleĠlabelsĠforĠtheĠpointĠ(2000,Ġ2024
Rollout 67, Act: -2.190
pĠ+Ġ2q)Ġ=Ġ1848.Ġ...(3)ĊĊSoĠnowĠweĠhaveĠthree
Rollout 67, Act: -2.307
\]Ċ\[Ċc_8Ġ=Ġ-72Ġ\cdotĠ64Ġ+Ġ393

DOWN_PROJ

Top 16 Positive Activations
Rollout 96, Act: 2.568
)Ġ=Ġ(1740/169,Ġ-1134/169
Rollout 81, Act: 2.387
/17:ĊĊtan(theta)Ġ=Ġ|(-h/17Ġ-Ġh/3)/(1Ġ+
Rollout 78, Act: 2.328
.ĠTherefore,ĠtheirĠrelativeĠspeedĠisĠ620/9Ġ+Ġ20Ġ=Ġ62
Rollout 22, Act: 2.308
Ġv_meanĠ=Ġsqrt(18,168.7768Ġ/Ġ0.0
Rollout 26, Act: 2.293
Ġ=Ġ-3.ĊĊÎĶHĠ=Ġ-296ĠkJĠ+Ġ(-3)Ġ*Ġ(8.
Rollout 37, Act: 2.292
ĠwouldĠbeĠa1^{k1}a2^{k2}Ġ-Ġa1^{k2}
Rollout 36, Act: 2.270
syĠ+Ġs2)Ġ=Ġ36000Ġ-Ġ18000ĊS
Rollout 96, Act: 2.255
_y)Ġ=Ġ(1740/169,Ġ-1134/16
Rollout 96, Act: 2.240
69.ĠSoĠOA_x=4056/169Ġ-1740/
Rollout 96, Act: 2.213
ĠfromĠO(1740/169,Ġ-1134/169
Rollout 36, Act: 2.190
Ġ(1'):ĊĊSĠ=Ġ36000Ġ+Ġ2sĠxĠ-Ġs2ĊĊ
Rollout 64, Act: 2.174
ĠbeĠpositive:ĊĊa1Ġ=ĠAĠ+Ġ2BĠ>0.ĠSimilarly,Ġa2Ġ=AĠ+
Rollout 36, Act: 2.169
00/s)ĠxĠ=Ġ(18000/sĠ+Ġ7200/s)
Rollout 67, Act: 2.151
ButĠfromĠequationĠ1:ĠpĠ=1440Ġ/Ġa.ĊĊSimilarly,ĠfromĠequationĠ2
Rollout 79, Act: 2.143
8,4)Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠcrosswordĠclue.ĠTheĠclue
Rollout 5, Act: 2.137
5HĊ2.Ġ(xĠ+Ġ175)(HĠ-Ġh)Ġ=Ġ12
Top 16 Negative Activations
Rollout 48, Act: -6.184
ĠsoĠit'sĠperhapsĠeasierĠtoĠsumĠeverythingĠinĠ32Ġbits,ĠthenĠaddĠtheĠcarryoverĠbitsĠafterward
Rollout 82, Act: -6.185
Ġsymbols.ĠThus,ĠallĠcodewordsĠhaveĠlengthĠ2.ĠThen,ĠtheĠexpectedĠnumberĠofĠsymbolsĠwould
Rollout 97, Act: -6.199
ĠasĠwrittenĠisĠ2N2O5Ġ->Ġ2N2O4}+O2}
Rollout 38, Act: -6.200
Ġ(LĠ-Ġx)Ġ(east)Ġ]Ġ/Ġ8.6Ġ=Ġ[2LĠ-ĠxĠ+
Rollout 35, Act: -6.213
ĠD=âĪĴ10,ĠwhereĠ(nâĪĴ17)(nâĪĴ24)=âĪĴ10
Rollout 19, Act: -6.236
.ĊĊ-ĠThenĠsquaredĠgivesĠ100âī¡17modk.ĊĊ-ĠThusĠkĠdivides83
Rollout 89, Act: -6.255
frac{2}{5}Ġ\timesĠ\frac{500}{3}Ġ=Ġ\frac{
Rollout 90, Act: -6.313
0ĠmodĠ2018:ĠN*10^4Ġ+ĠM*10^3
Rollout 24, Act: -6.352
.ĊĊButĠevenĠifĠ|R|=|C|Ġdoesn'tĠnecessarilyĠgiveĠsameĠrowĠnumberĠbetweenĠdifferentĠpairs.ĊĊ
Rollout 38, Act: -6.391
ĠmetersĠtoĠtheĠpointĠ(-Ġ(LĠ-Ġx),Ġ0),ĠthenĠsouthĠWĠmetersĠtoĠ(-Ġ(L
Rollout 21, Act: -6.410
ĠwouldĠcontainĠallĠinteger-valuedĠperiodicĠfunctionsĠwithĠperiodĠ1ĠandĠh(0)=0ĠdueĠtoĠf
Rollout 83, Act: -6.501
ĠarrangementsĠfixedĠexceptĠidentity.ĊĊSameĠreasoningĠforĠrotationĠbyĠ6.ĊĊForĠrotationĠby1ĠpositionĠ-ĠitĠaffects
Rollout 53, Act: -6.589
Ġ=Ġ2.5ĠwhichĠlinksĠtoĠACĠ=Ġ5.ĠSoĠthreeĠpieces:ĠCEĠ=Ġ4
Rollout 21, Act: -6.788
ĠSoĠfĠwouldĠhaveĠtoĠbeĠaĠfunctionĠwithĠperiodĠ1.ĠThen,ĠinĠcombinationĠwithĠotherĠequations,
Rollout 14, Act: -6.942
ĠdxĠ)Ġ*Ġsqrt(âĪ«A^inftyĠ1ĠdxĠ)=Ġsqrt(J2)*Ġsqrt(in
Rollout 29, Act: -7.189
ĠcompartmentĠexertsĠhigherĠpressureĠinĠthatĠcompartment,ĠbutĠdoesn'tĠaffectĠtheĠopenĠcompartment'sĠpressure.ĠIfĠthey

Layer 44

GATE_PROJ

Top 16 Positive Activations
Rollout 99, Act: 2.272
.5Ġ\),ĠandĠ\(Ġx_2Ġ=Ġ1Ġ\),ĠweĠneedĠtoĠperformĠtwoĠiterationsĠof
Rollout 19, Act: 2.248
^3Ġ\)ĠisĠdividedĠbyĠ\(ĠnĠ+Ġ5Ġ\),ĠtheĠremainderĠisĠ17.
Rollout 30, Act: 2.108
Ġ-Ġ1$ĠforĠ$nĠ\geqĠ2$Ċ-Ġ$a_1Ġ=Ġa_
Rollout 2, Act: 2.086
ĠweĠconsiderĠtheĠproductĠmoduloĠ8ĠandĠmoduloĠ125Ġseparately,ĠthenĠcombineĠtheĠresultsĠusingĠthe
Rollout 55, Act: 2.079
3}}{2}Ġ&Ġ0Ġ&Ġ-\frac{1}{2}Ġ\\Ġ0Ġ&
Rollout 37, Act: 2.036
Ġ&Ġa_{2015}^{k_2}Ġ&Ġ\cdotsĠ&Ġa_{2
Rollout 59, Act: 2.036
ĠprobabilityĠthatĠtheyĠgetĠtheĠsameĠnumberĠofĠheads.ĊĊFirst,ĠweĠdetermineĠtheĠprobabilityĠdistributionĠforĠtheĠnumberĠof
Rollout 50, Act: 2.014
,ĠandĠthenĠcomputeĠ\(mĠ+Ġn\).ĊĊFirst,ĠletĠ\(yĠ=Ġ\log_{20
Rollout 82, Act: 2.001
9},Ġ\frac{8}{69}Ġ\right)ĊĠĠĠ\]ĊĊ2.Ġ**Grouping
Rollout 78, Act: 1.999
ĠfindĠtheĠlengthĠofĠeachĠtrain.ĊĊKeyĠsteps:Ċ1.ĠLetĠ\(ĠLĠ\)ĠbeĠtheĠlengthĠof
Rollout 32, Act: 1.996
{M}^{2009}\).ĊĊFirst,ĠweĠnoteĠthatĠ\(\omega\)ĠisĠa
Rollout 36, Act: 1.996
\sqrt{10}\),Ġ\(CPĠ=Ġ60\sqrt{5}\),Ġ\(DP
Rollout 88, Act: 1.991
ĠofĠweightsĠ(excludingĠbiases)ĠinĠthisĠlayer:ĊĊ1.ĠEachĠfilterĠhasĠdimensionsĠ5x5x
Rollout 95, Act: 1.989
{671}k}{3Ġ\timesĠ2450}\right)\).ĊĠĠĠ-
Rollout 95, Act: 1.979
Ġ=ĠQ_0,ĠQ_1,ĠQ_2,Ġ\ldots,ĠQ_{24
Rollout 29, Act: 1.958
ĠPĠ\)ĠatĠtheĠbottomĠofĠtheĠjug.ĊĊ1.Ġ**UnderstandingĠGaugeĠPressure**:ĠGaugeĠpressureĠisĠthe
Top 16 Negative Activations
Rollout 59, Act: -3.373
2Ġ=Ġ16*24.5?ĠWait,Ġ16Ġ*24.5
Rollout 70, Act: -3.380
ĠSoĠdigitsĠare1Ġ(forĠ350?),ĠwaitĠno:ĊĊWaitĠ363ĠdividedĠby
Rollout 97, Act: -3.381
85/0.0005=ĠapproxĠ4605.17Ġseconds.ĊĊ
Rollout 71, Act: -3.386
Ġ<=Ġ8044/12ĠâīĪĠ670.333,Ġso
Rollout 14, Act: -3.388
)^{1/2Ġ}Ġ)^{.1Ġ}Ġ+Ġsqrt(J)/2.ĊĊĠWait,ĠactualĠcalculation
Rollout 69, Act: -3.404
(theta_0)/cos(theta_{-1})Ġ?ĊĊButĠtheta_{-1}ĠisĠundefined.ĠH
Rollout 54, Act: -3.416
Ġweaker,ĠthatĠmeansĠHSâĪĴĠisĠaĠstrongerĠbase.ĠWait,Ġwait.ĠNo:Ġwait,ĠtheĠlower
Rollout 69, Act: -3.421
cos(3Ġtheta'Ġ/3^j)Ġ?ĊĊWait,ĠperhapsĠconfusedĠhere.ĠLet'sĠmakeĠsubstitution
Rollout 26, Act: -3.422
3.5ĠkJĠ(fourĠsigĠfigs?)ĊĊThen,ĠtheĠadjustmentĠforĠHg(l)Ġis
Rollout 30, Act: -3.428
Ġ/Ġ(a_{n+1Ġ}Ġ)2Ġ)ĊĊWait,Ġno:ĊĊWait,ĠifĠa_{n
Rollout 8, Act: -3.438
Ġ/Ġ(5Ġ-Ġ(5Ġ-r))!Ġ;Ġwait,ĠmaybeĠit'sĠbetterĠtoĠthink:ĊĊWait
Rollout 55, Act: -3.540
3\*2018pi/6.ĠWaitĠmaybeĠthat?ĊĊWait,ĠtheĠrotationĠmatrixĠfor
Rollout 69, Act: -3.543
1}\timesĠ10ĠtoĠ3Ġscaling.ĊĊAlternatively,ĠmaybeĠconsiderĠtelescopingĠthroughĠtrigonometric
Rollout 83, Act: -3.564
=1296/s=128,ĠwhichĠisĠ1296/9=
Rollout 70, Act: -3.565
9:ĠN=331Ġ+16=331?ĠWait,ĠwaitĠN=3
Rollout 75, Act: -3.749
5+27+12=54+.ĠWait,Ġno.ĊĊWait,Ġhonestly,ĠseemsĠlike

UP_PROJ

Top 16 Positive Activations
Rollout 89, Act: 1.365
33=66.666.ĊĊHmm,ĠsoĠ66.6Ġrecurring.ĊĊIn
Rollout 36, Act: 1.351
*1,856=18,560,000ĊĊ1,8
Rollout 64, Act: 1.342
1000ĊĊn=3:Ġ-998Ġ+3996=29
Rollout 81, Act: 1.294
3,ĠdeltaĠy=Ġ0Ġ-Ġh=Ġ-h.Ġhence,Ġslope=ĠdeltaĠy/delta
Rollout 23, Act: 1.270
1.925e32ĠBq.ĊĊButĠagainĠnotĠtheĠanswerĠoptions.ĊĊAlternatively,Ġgiven
Rollout 82, Act: 1.251
4)/69âīĪ2.087.ĊĊĠsameĠresult.ĠThereforeĠhigherĠthan2.ĊĊHence
Rollout 62, Act: 1.247
=2.ĠNope.ĠcheckĠifĠprimeĠMaybe.ĊĊOKĠperhapsĠbackĠtoĠapproach.ĊĊTurnsĠout,ĠI
Rollout 92, Act: 1.228
Ġ61.5ĠcmĠfromĠtheĠlens.ĠSoĠpreviousĠcalculationĠisĠcorrect.ĊĊIĠthinkĠtheĠanswerĠis
Rollout 75, Act: 1.214
Ġfour:Ġ4.ĠTotalĠsumĠ9.ĠHmm,Ġno.ĠMaybeĠsplittingĠintoĠnumbersĠwhere?ĊĊWait
Rollout 75, Act: 1.204
,ĠunlessĠ2mĠ-1ĠisĠsquare.ĊĊHmm,ĠperhapsĠtheĠidentityĠisĠthis:ĊĊm2Ġ=
Rollout 36, Act: 1.202
00*1,856=18,560,000ĊĊ1
Rollout 22, Act: 1.186
)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠtoĠcalculateĠtheĠmeanĠmolecular
Rollout 93, Act: 1.172
ĠalsoĠbeĠinterpretedĠasĠ(probabilityĠamplitudes?)ĠHmm.ĠAlternatively,ĠmaybeĠweĠcanĠuseĠtheĠidentityĠthat
Rollout 3, Act: 1.168
)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculateĠtheĠmeanĠmolecularĠspeed
Rollout 77, Act: 1.159
Ġproperties:Ġ\[f(x,Ġx)Ġ=Ġx,\;Ġf(x,Ġy)Ġ=Ġf(y
Rollout 55, Act: 1.159
Ġ\renewcommand{\arraystretch}{1.5}Ġ\begin{pmatrix}Ġ\frac{\
Top 16 Negative Activations
Rollout 31, Act: -2.313
0Ġ=>ĠxĠyĠ=Ġ-1ĊĊSo,ĠsubstituteĠyĠfromĠtheĠfirstĠequationĠintoĠtheĠsecondĠequation:ĊĊ
Rollout 51, Act: -2.322
x|ĠdxĊĊAlternatively,ĠifĠIĠletĠtĠ=ĠxĠ-ĠÏĢĠforĠtheĠsecondĠintegral,Ġsubstitute:ĊĊ
Rollout 83, Act: -2.331
44.ĊĊAlternatively,ĠtoĠverifyĠconsiderĠspecificĠsmallĠcase,ĠlikeĠtotalĠacceptableĠarrangementsĠwithoutĠsymmetry=12
Rollout 40, Act: -2.331
1421356.ĊĊSoĠtheĠgreatestĠintegerĠlessĠthanĠorĠequalĠtoĠ19.
Rollout 38, Act: -2.338
2857.ĠLet'sĠdivideĠnumeratorĠandĠdenominatorĠbyĠ48:Ġ2304
Rollout 34, Act: -2.343
ĠtheĠinitialĠcondition.ĠFromĠtheĠoriginalĠintegralĠequation,ĠwhenĠtĠ=Ġ0:ĊĊx(0)Ġ=
Rollout 12, Act: -2.355
ĠvanishĠorĠsomething.ĊĊAlternatively,ĠmaybeĠweĠcanĠsubstituteĠvariablesĠtoĠabsorbĠtheĠdependenceĠonĠν?ĊĊAlternatively,Ġinstead
Rollout 12, Act: -2.357
,ĠmaybeĠweĠcanĠperformĠaĠsubstitutionĠinĠtheĠintegralĠtoĠmakeĠtheĠintegralĠtransformĠintoĠanotherĠformĠwhereĠtheĠdependence
Rollout 0, Act: -2.361
ĠLetĠmeĠcheck.ĠTheĠprimesĠlessĠthanĠorĠequalĠtoĠ20Ġare:Ġ2,Ġ3
Rollout 5, Act: -2.379
Ġ=Ġ181.25,ĠgreatestĠintegerĠnotĠexceedingĠ181.25Ġis
Rollout 11, Act: -2.401
Ġ+Ġc/a.ĠSoĠlet'sĠexpressĠqĠinĠtermsĠofĠa,Ġb,Ġc.ĠSinceĠabcĠ=
Rollout 30, Act: -2.446
}Ġb_nĠ-1.ĊĊBecauseĠwhenĠnĠisĠreplacedĠbyĠn+1ĠinĠtheĠb_{n+
Rollout 12, Act: -2.510
ĠtheĠvariableĠofĠintegration.ĠLetĠmeĠconsiderĠswitchingĠtheĠvariableĠtoĠt,ĠbutĠnotĠsure.ĊĊWait,Ġlet
Rollout 58, Act: -2.521
Ġ=Ġ2018.ĠTherefore,ĠtheĠnumberĠofĠpartsĠisĠ2018.ĠTherefore
Rollout 7, Act: -2.639
ĠisĠ56.ĠThen,ĠdueĠtoĠanĠerror,ĠallĠscoresĠareĠincreasedĠbyĠ5.ĠAfter
Rollout 51, Act: -2.642
ĠcanĠmakeĠsubstitutionĠtoĠsimplifyĠthings.ĠMaybeĠshiftingĠtheĠvariableĠtoĠeliminateĠθ?ĠLet'sĠletĠyĠ=

DOWN_PROJ

Top 16 Positive Activations
Rollout 70, Act: 2.746
306.ĠWaitĠthat'sĠcorrect.ĠSoĠwhyĠearlierĠwhenĠIĠtriedĠtoĠdivideĠ153
Rollout 73, Act: 2.660
+3=23.ĠWait,Ġno,ĠwhyĠ+3?ĠBecauseĠsum_{j=1}
Rollout 6, Act: 2.629
ĠifĠtheĠadviceĠisĠgeneratedĠinĠpolynomialĠtime,ĠthenĠwhyĠnotĠjustĠincorporateĠthatĠpolynomial-timeĠcomputationĠintoĠtheĠmain
Rollout 15, Act: 2.576
ĠfromĠallĠon,ĠendsĠafterĠtwoĠsteps.ĠThenĠwhyĠdidĠIĠearlierĠthoughtĠitĠcycles?ĊĊBecauseĠIĠmade
Rollout 99, Act: 2.567
ĠfirstĠiteration.ĠTheĠquestionĠisĠambiguouslyĠworded.ĊĊHowever,ĠgivenĠthatĠitĠspecificallyĠasksĠforĠ"the
Rollout 62, Act: 2.566
847,ĠsoĠm=(1+34)/2=17.5~Ġdecimal.
Rollout 97, Act: 2.554
ĠWhichĠgivesĠYĠ~2.3.ĊĊButĠthenĠwhyĠwasĠtheĠpreviousĠanswerĠapproxĠ4.6?ĊĊ
Rollout 9, Act: 2.547
ĠconfirmĠtheĠpositions.ĠIfĠtheĠsquareĠisĠlabeledĠAIME,ĠmovingĠaroundĠtheĠsquare,ĠthenĠmaybeĠAĠis
Rollout 7, Act: 2.540
ĠitselfĠisn'tĠchanged.ĠHmm.ĠThisĠisĠambiguous.ĊĊButĠaccordingĠtoĠtheĠproblemĠstatement:Ġ"dueĠto
Rollout 68, Act: 2.538
ĠotherĠisĠnot.ĠSoĠtwoĠcases:ĊĊCaseĠ1:ĠTheĠquadraticĠequationĠhasĠexactlyĠoneĠrealĠsolutionĠwhich
Rollout 76, Act: 2.534
3,ĠwaitĠ417:Ġ4+1+7=12,ĠsoĠdivisibleĠby
Rollout 48, Act: 2.533
ĠweĠshouldĠexpressĠtheĠnumbersĠas:ĊĊ0xĠEĠĠ1ĠĠ9ĠĠĠEĊĊ+0
Rollout 62, Act: 2.521
ĠsoĠ2081-2080=1.ĠNotĠdivisible.ĠByĠ7=
Rollout 57, Act: 2.509
ĠButĠthenĠCĠmustĠbeĠopenĠinĠtheĠdensityĠtopology.ĊĊThisĠleadsĠtoĠtheĠconclusionĠthatĠinĠtheĠdensityĠtopology
Rollout 95, Act: 2.500
inĠliesĠtheĠproblem.ĠWhichĠisĠtheĠcorrectĠrelationship?ĊĊHmm.ĠIfĠtriangleĠAPkQkĠisĠsimilar
Rollout 1, Act: 2.499
ĠfindĠsuchĠaĠyĠbyĠfittingĠitĠaccordingly.ĊĊSoĠwhyĠinĠtheĠpreviousĠexampleĠwithĠtranslatedĠsetĠS',Ġit
Top 16 Negative Activations
Rollout 62, Act: -2.264
Ġ-5Ġ+4=24âī¡3âī¡3âīł0ĊĊm=6:36Ġ-
Rollout 62, Act: -2.268
Ġ-6Ġ+4=34âī¡6âī¡6âīł0ĊĊNoĠsolution.ĠTherefore,ĠmodulusĠ
Rollout 0, Act: -2.270
}{b}\)ĠandĠcalculateĠtheĠproductĠofĠtheĠresultingĠnumeratorĠandĠdenominator.ĠWeĠneedĠtoĠdetermineĠhowĠmany
Rollout 64, Act: -2.275
2009}Ġ+a1Ġ-kĠâī¤0ĊĊButĠthisĠseemsĠnotĠhelpful.ĠAlternatively,
Rollout 49, Act: -2.290
-finiteĠopenĠcoverĠofĠXĠhasĠaĠcountableĠsubcover.ĠWait,ĠbutĠnotĠsure.ĊĊAlternatively,
Rollout 50, Act: -2.322
ĠtermsĠtoĠtheĠleft-handĠside:ĊĊ\[Ċt(\log_{10}22Ġ+Ġ\log
Rollout 1, Act: -2.337
2Ġ-Ġ2cĠ+Ġd2)/2Ġ=Ġ0ĠâĩĴĠ0Ġ=Ġ0.ĊĊThus
Rollout 10, Act: -2.344
2,...ĊĊDifferenceĠbetweenĠ0ĠandĠ2:Ġ2ĊĊ2ĠtoĠ5:Ġ3ĊĊ
Rollout 82, Act: -2.356
)/69Ġ=Ġ(16+15=31;Ġ31+24=
Rollout 58, Act: -2.358
ĠoneĠquestionĠcorrectĠinĠtheĠentireĠexamĠversusĠgettingĠnoneĠcorrect.ĠThatĠseemsĠlikeĠweĠhaveĠtoĠcomputeĠtheseĠtwo
Rollout 41, Act: -2.365
ĠnumberĠlikeĠ123,ĠwhichĠhasĠthreeĠdifferentĠdigits,ĠwouldĠnotĠbeĠcounted.ĠGotĠit.ĊĊ
Rollout 75, Act: -2.368
9ĠsplitĠintoĠ4+4+1ĠgivesĠthreeĠmoreĠsquares,ĠandĠtheĠ+1ĠisĠalreadyĠcovered
Rollout 59, Act: -2.381
1+16.ĊĊ9+100=109;Ġ109+1
Rollout 20, Act: -2.390
3,47.ĊĊSumĠâĨĴ2+3=5;Ġ5+7=12;
Rollout 41, Act: -2.484
ĠandĠ2),ĠbutĠ123ĠhasĠthreeĠdigits.ĠSoĠweĠneedĠtoĠcountĠallĠthree-digit
Rollout 21, Act: -2.531
Ġf(x)(f(x)Ġ-Ġ1)Ġ=Ġ0,ĠwhichĠforcesĠf(x)ĠtoĠbe

Layer 45

GATE_PROJ

Top 16 Positive Activations
Rollout 40, Act: 3.606
6ĠisĠ19.ĠTherefore,ĠtheĠanswerĠisĠ19.ĊĊButĠletĠmeĠverifyĠmyĠsteps
Rollout 27, Act: 3.588
.ĠGivenĠthisĠconfusion,ĠperhapsĠtheĠinitialĠstepĠisĠcorrectĠwhereĠÎĶE_rmsĠisĠ(1/2
Rollout 0, Act: 3.551
,ĠcorrespondingĠtoĠ\boxed{128}?ĠĠOrĠwait,ĠwaitĠprimeĠfactorsĠofĠ20
Rollout 2, Act: 3.527
ĠtheĠremainderĠis109.ĊĊTherefore,Ġanswer:109.ĊĊButĠletĠmeĠrecheckĠall
Rollout 4, Act: 3.517
ĠapproximatelyĠ0.88%.ĊĊWait,ĠmaybeĠ0.88ĠisĠokay.ĠAlternatively,Ġlet
Rollout 87, Act: 3.515
]Ġ}ĊĊSinceĠIĠrecallĠthisĠfromĠinternetĠsources.ĠOrĠitĠmightĠbeĠexpressedĠusingĠdifferentĠconstants.ĊĊAlternatively,
Rollout 99, Act: 3.508
714012ĊĊSoĠx3âīĪ2.15733466
Rollout 3, Act: 3.506
168ĠorĠ169.ĠMaybeĠ168Ġm/s.ĊĊButĠcheckingĠfurtherĠon
Rollout 64, Act: 3.487
Ġ2^{2011})Ġ?ĠOrĠno,ĠmyĠpriorĠcalculationsĠyieldedĠthatĠtheĠtotalĠnumberĠis
Rollout 13, Act: 3.486
616ĊĊTherefore,ĠA_annualĠâīĪĠ$9,441.12.
Rollout 38, Act: 3.471
24=61.ĊĊTherefore,ĠtheĠanswerĠisĠ\boxed{61}ãĢĤĊĊ**FinalĠAnswer
Rollout 5, Act: 3.463
ĠisĠ181.ĠTherefore,ĠtheĠanswerĠisĠ181.ĠĊĊWait,ĠbutĠlet
Rollout 60, Act: 3.460
6.00ĊĊSoĠtotalĠf''ĠâīĪĠ492Ġ+Ġ6Ġ=Ġ4
Rollout 33, Act: 3.460
ĠcorrectĠanswer.ĠHence,ĠI'dĠconcludeĠtheĠanswerĠis\boxed{461}.ĊĊ**Final
Rollout 87, Act: 3.434
.ĠConsideringĠtheĠoptions,ĠIĠthinkĠtheĠanswerĠisĠlikely:ĊĊVĠ=Ġ\boxed{Ġ\dfrac{\
Rollout 5, Act: 3.411
ĠisĠ181.ĠTherefore,ĠfinalĠanswerĠisĠ181.ĠĊĊIĠthinkĠthat's
Top 16 Negative Activations
Rollout 39, Act: -1.167
Ġ\sum_{k=1}^{321}Ġ1Ċ\]ĊĊTheĠfirstĠsumĠis
Rollout 37, Act: -1.189
}Ġ&Ġ\cdotsĠ&Ġa_2^{k_{2015}}Ġ\\Ċ\vdots
Rollout 37, Act: -1.195
2015}}Ġ\\Ċa_2^{k_1}Ġ&Ġa_2^{k_
Rollout 41, Act: -1.197
10,000ĠhaveĠatĠmostĠtwoĠdifferentĠdigits,ĠweĠconsiderĠnumbersĠofĠdifferentĠlengthsĠ(1
Rollout 73, Act: -1.200
left(Ġ\sum_{i=0}^{j-1}Ġ2^jĠ-Ġ\sum
Rollout 73, Act: -1.216
0Ġ\)ĠtoĠ\(Ġ2^{10}Ġ\).ĠWeĠneedĠtoĠconsiderĠallĠpairsĠ\((
Rollout 53, Act: -1.221
((-a,Ġ0)\)ĠandĠ\((a,Ġ0)\),ĠandĠpointĠ\(C\)
Rollout 39, Act: -1.275
}Ġ10^kĠ-Ġ\sum_{k=1}^{321}Ġ1
Rollout 73, Act: -1.276
,Ġ\ldots,Ġ2^{10}\}Ġ\),ĠweĠneedĠtoĠfindĠtheĠsumĠof
Rollout 39, Act: -1.289
^{k-1}Ġ+Ġ10^{k-2}Ġ+Ġ...Ġ+Ġ10^
Rollout 37, Act: -1.295
15}^{k_{2015}}Ċ\end{pmatrix}Ċ\]ĊĊ###ĠKey
Rollout 99, Act: -1.314
x)Ġ-ĠxĠ+Ġ3Ġ=Ġ0Ġ\)ĠwithĠinitialĠpointsĠ\(Ġx_0Ġ=Ġ
Rollout 82, Act: -1.344
0,Ġ\(Ġ\frac{8}{69}Ġ\)ĠisĠencodedĠasĠ81ĊĠĠĠĠ
Rollout 61, Act: -1.351
Ġ\(Ġ2^kĠ+Ġ2^mĠ\)Ġ(withĠ\(ĠkĠ>ĠmĠ\))Ġis
Rollout 37, Act: -1.361
^{k_1}ĠĠa_2^{k_2}ĠĠ...ĠĠa_2^{
Rollout 73, Act: -1.553
jĠ-Ġ\sum_{i=0}^{j-1}Ġ2^iĠ\right)Ċ

UP_PROJ

Top 16 Positive Activations
Rollout 7, Act: 2.925
Ġ"dueĠtoĠanĠerrorĠinĠtheĠwordingĠofĠaĠquestion,ĠallĠscoresĠareĠincreasedĠbyĠ5.ĠAt
Rollout 9, Act: 2.806
Ġsquare.ĠThenĠtheĠtriangleĠGEMĠoverlapsĠtheĠsquare,ĠandĠtheirĠcommonĠareaĠisĠ80ĠsquareĠunits
Rollout 7, Act: 2.728
:ĠTheseĠareĠparticipantsĠwithĠoriginalĠscoreĠ<65,ĠaverageĠ56.ĠTheirĠtotalĠscoreĠisĠ
Rollout 7, Act: 2.722
Ġ"dueĠtoĠanĠerrorĠinĠtheĠwordingĠofĠaĠquestion,ĠallĠscoresĠareĠincreasedĠbyĠ5.ĠAt
Rollout 7, Act: 2.692
ĠatĠ65.ĠWait,ĠtheĠscoresĠareĠincreasedĠbyĠ5,ĠsoĠoriginalĠscoresĠbecomeĠoriginalĠ+
Rollout 15, Act: 2.674
ĠruleĠdoesn'tĠmentionĠtheĠcurrentĠstate.ĠTheĠnextĠstateĠdependsĠonlyĠonĠtheĠnumberĠofĠneighbors.ĠSoĠregardless
Rollout 7, Act: 2.665
,Ġ6Ġremaining.ĠOriginalĠaverageĠofĠ2Ġstudents=62,ĠsoĠtotalĠscore=62
Rollout 33, Act: 2.658
ĠmanyĠsuchĠpermutationsĠareĠthere?ĠForĠtheĠfirstĠthreeĠpositions:ĊĊ-ĠFirstĠtwoĠpositions:Ġ2!Ġarrangements
Rollout 7, Act: 2.658
"dueĠtoĠanĠerrorĠinĠtheĠwordingĠofĠaĠquestion,ĠallĠscoresĠareĠincreasedĠbyĠ5.ĠAt
Rollout 25, Act: 2.657
ĠTheĠproblemĠstatesĠthatĠwhenĠtheĠasteroid'sĠmassĠchanges,ĠtheĠparametersĠvĠ(speedĠatĠinfinity)ĠandĠr
Rollout 8, Act: 2.637
ĠarrangedĠinĠsameĠcolorĠorder,ĠbutĠdifferentĠactualĠcards,Ġit'sĠstillĠcountsĠasĠtheĠsameĠcolorĠsequence.
Rollout 90, Act: 2.634
ĠsaysĠthatĠwhenĠyouĠremoveĠtheĠmiddleĠdigitĠfromĠA,ĠyouĠgetĠB.ĠForĠAĠwithĠanĠoddĠnumber
Rollout 58, Act: 2.626
ĠmultiplyĠacrossĠquestions.ĊĊButĠtheĠproblemĠisĠaboutĠtheĠprobabilityĠofĠgettingĠexactlyĠoneĠquestionĠcorrectĠinĠtheĠentireĠexam
Rollout 59, Act: 2.624
(X=k)]^2,ĠsoĠtotalĠprobabilityĠisĠsumĠoverĠkĠ[P(X=k)]2.ĊĊSo
Rollout 32, Act: 2.623
,ĠandĠthenĠsummingĠallĠthoseĠtotalsĠtogether.ĠSoĠessentially,ĠifĠSĠisĠtheĠsumĠofĠmatricesĠM
Rollout 89, Act: 2.621
ĠtheĠquestionĠisĠsaying.ĠItĠsaysĠthatĠinĠaĠclass,Ġ2/5ĠofĠtheĠchildrenĠareĠgirls
Top 16 Negative Activations
Rollout 79, Act: -1.682
7,ĠnotĠ8+4.ĠNotĠmatching.ĊĊAlternatively,Ġ"MachineĠOrgans".ĠM,ĠA
Rollout 49, Act: -1.689
able.ĠTherefore,ĠitĠisĠaĠcounterexample.ĊĊHence,ĠXĠmayĠstillĠresistĠbeingĠsecond-countable
Rollout 79, Act: -1.720
Ġisn'tĠrelated.ĠLettersĠdon'tĠfit.ĠThisĠisn'tĠworking.ĠPerhapsĠI'mĠmissingĠaĠcommonĠphrase
Rollout 49, Act: -1.722
ĠButĠweĠprovedĠXĠisĠfirst-countable,ĠbutĠdonâĢĻtĠknownĠifĠsecond-countable.ĊĊButĠXĠis
Rollout 79, Act: -1.724
ĠbutĠno.ĠAsĠbefore.ĠWhat'sĠanother?ĠLet'sĠtryĠ"HarmonizingĠCES".ĠNotĠthat
Rollout 79, Act: -1.725
ĠâĢĵĠbutĠthat'sĠ10Ġletters.ĠHmm.ĊĊAlternatively,ĠperhapsĠ"MinorĠchanges"ĠcanĠbeĠrearr
Rollout 79, Act: -1.729
ĠweĠhaveĠonlyĠoneĠE,Ġno.ĠSoĠno.ĊĊAnotherĠguess:Ġ"AnsweringĠMachine".ĠButĠthat
Rollout 11, Act: -1.736
.ĠNoĠgood.ĊĊAlternatively,ĠtakeĠaĠ=Ġ\overline{1/b},ĠleadingĠtoĠmodulusĠ|a
Rollout 79, Act: -1.749
zineĠà¤ļमà¥ĩ"ĠâĢĵĠnope.ĊĊWait,ĠmaybeĠ"Hormones"ĠâĢĶĠusing
Rollout 21, Act: -1.755
,Ġetc.,ĠbutĠconstantsĠareĠlikeĠ1,Ġ2,ĠÏĢ,Ġetc.ĊĊWait,ĠsupposeĠwe
Rollout 79, Act: -1.761
.ĊĊAlternatively,Ġ"AgencyĠOffice"ĠâĢĵĠbutĠlettersĠdon'tĠmatch.ĊĊWait,ĠperhapsĠ"Inchoate
Rollout 79, Act: -1.770
12Ġletters.ĠSoĠSearchĠWingĠisn'tĠcorrect.ĊĊThenĠIĠhitĠuponĠtheĠidea:ĠmaybeĠ"Main
Rollout 49, Act: -1.790
.ĠForĠexample,ĠinĠaĠlocallyĠcompactĠHausdorffĠÏĥ-compactĠspace,ĠyouĠcanĠshowĠit
Rollout 79, Act: -1.825
ĠanagramĠofĠ"minorĠchanges":ĠM,ĠI,ĠN,ĠO,ĠR,ĠC,ĠH,
Rollout 79, Act: -1.880
'sĠsee.ĠMaybeĠ"Chroning"?ĠButĠweĠdon'tĠhaveĠaĠT.Ġ"Horming"?
Rollout 79, Act: -1.899
ĠtheĠanswerĠisĠ"Monitoring...Ġsomething".ĠButĠlettersĠaren'tĠmatching.ĠAlternatively,Ġ"ChangeĠMinor"ĠâĢĵ

DOWN_PROJ

Top 16 Positive Activations
Rollout 14, Act: 2.641
1/3}Ġ\right)^{2/Ġ(3/2Ġ-1)}Ġ}Ġ\right)^
Rollout 11, Act: 2.548
ateĠvariables?ĠNotĠsure.ĠMaybeĠifĠ|a|ĠâīłĠ1,ĠbutĠsinceĠabcĠ=Ġ1
Rollout 81, Act: 2.536
ĠCABĠisĠtheĠangleĠatĠpointĠAĠbetweenĠpointsĠC,ĠA,ĠandĠB.ĠWait,ĠmaybeĠthisĠcoordinate
Rollout 78, Act: 2.505
1/13.5?ĠWait,Ġno.ĊĊWait,Ġ440ĠdividedĠbyĠ9
Rollout 38, Act: 2.503
04/672=2304Ġ÷Ġ672=ĠapproxĠ3
Rollout 60, Act: 2.489
}$.ĠTheĠsoundĠisĠreflectedĠbyĠaĠlargeĠcarĠapproachingĠtheĠsourceĠwithĠaĠspeedĠofĠ$2Ġ\mathrm
Rollout 28, Act: 2.454
,ĠtheĠmidpointĠofĠOG?ĠOrĠrelatedĠtoĠmedians.ĊĊWait,ĠinĠtheĠgeneralĠcoordinateĠsystem,ĠG
Rollout 87, Act: 2.421
Ġ0ĠtoĠaĠ/ĠcosĠθĠorĠaĠ/ĠsinĠθ,ĠdependingĠonĠtheĠangle,Ġwhich
Rollout 32, Act: 2.391
2Ġ+0Ġ+Ġ(âĪĴÏī2ĠfromĠS)Ġ??ĠWaitĠno.ĠLetĠmeĠrecomputetheĠprevious
Rollout 49, Act: 2.382
pace.ĠButĠifĠXĠisĠuncountableĠandĠregularĠ(sinceĠHausdorffĠandĠhereditarily
Rollout 38, Act: 2.382
,ĠbutĠpointĠDĠisĠnotĠnecessarilyĠatĠtheĠsouthwestĠcorner.ĠIt'sĠsomewhereĠonĠtheĠsouthĠedge,Ġmaybe
Rollout 61, Act: 2.379
ĠdifferentĠpairsĠ(m1,Ġk1)ĠandĠ(m2,Ġk2),ĠevenĠifĠd=k
Rollout 54, Act: 2.362
.ĠWait,ĠtheĠpKaĠofĠcyclohexanolĠisĠaroundĠ19-20Ġ(where
Rollout 71, Act: 2.352
105),Ġ(-3,-35),Ġ(-5,-21),Ġ(-7,-15
Rollout 26, Act: 2.350
Ġproblem,ĠallĠdataĠgivenĠseemsĠtoĠbeĠpreciseĠtoĠfourĠsignificantĠfigures.ĠForĠexample:ĊĊ-ĠHeatĠcapacityĠof
Rollout 8, Act: 2.339
Ġ9Ġ*Ġ8Ġ*Ġ7Ġ*Ġ6.ĠWait,ĠbecauseĠeachĠcardĠisĠdrawnĠwithoutĠreplacement
Top 16 Negative Activations
Rollout 6, Act: -2.461
ĠthisĠcase,ĠtheĠoracleĠTMĠcan,ĠgivenĠinputĠxĠofĠlengthĠn,ĠretrieveĠs_nĠfromĠtheĠoracle
Rollout 33, Act: -2.462
ĠthanĠkĠinĠtheĠfirstĠkĠpositions,ĠwhichĠisĠequivalentĠtoĠsayingĠthatĠtheĠfirstkĠpositionsĠcannotĠbeĠa
Rollout 1, Act: -2.474
ĠisĠd,ĠIĠneedĠtoĠfindĠaĠpointĠyĠsuchĠthatĠ{Ġsqrt(2)/dĠ(xĠ-
Rollout 93, Act: -2.480
4ÏĢ/1007Ġ=ĠÏĢ/2Ġ+ĠÏĢ/(2*1007).
Rollout 15, Act: -2.489
ĠoverĠGF(2).ĠTherefore,ĠtheĠoriginalĠproblemĠreducesĠtoĠthis.ĊĊHence,ĠTheĠanswerĠisĠthat
Rollout 19, Act: -2.522
7ĠmodkâĩĴ83âī¡0ĠmodkâĩĴkĠdivides83.ĠSince83ĠisĠprime
Rollout 70, Act: -2.524
(7)).ĊĊCalculatingĠthat:ĊĊ(ln(2/7))Ġ/Ġ(ln(10/
Rollout 22, Act: -2.540
62618âīĪ1.2206.ĠTotalĠofĠthat:Ġ81.
Rollout 19, Act: -2.550
ĠmodkâĩĴĠ83âī¡0modkâĩĴkĠdivides83.ĠTherefore,ĠpossibleĠk:
Rollout 86, Act: -2.562
)Ġ-Ġ(1/4)sin(pi)Ġ+Ġ(1/32)sin(2pi
Rollout 90, Act: -2.567
43,960Ġ+4,036=448,Ġ-Ġ44
Rollout 32, Act: -2.659
Ġ+669ĊĊTerm2ĠhasĠ+670ĠandĠ-670ÏīĊĊTerm3
Rollout 34, Act: -2.679
}Ġ+9Ġe^{s})Ġ=Ġ(4Ġ-12)Ġ+Ġ(-2Ġe^{-s}
Rollout 50, Act: -2.685
)ĊĊ=Ġ\((\log_{10}2)^2Ġ+Ġ\log_{10}1
Rollout 20, Act: -2.692
ĠcongruenceĠmodĠ2021ĠisĠequivalentĠtoĠtheĠcongruenceĠmodĠ43Ġand
Rollout 74, Act: -2.853
Ġ=ĠH_eĠ+ĠH_induced_NĠ=ĠH_eĠ[1Ġ+ĠK_bĠcos2θĠ-ĠK

Layer 46

GATE_PROJ

Top 16 Positive Activations
Rollout 52, Act: 3.182
\boxed{-2013}ĊĊ<|im_start|>answerĊToĠdetermineĠtheĠsmallestĠyummyĠintegerĠ\(ĠAĠ\),
Rollout 58, Act: 3.074
2037171}ĊĊ<|im_start|>answerĊGivenĠanĠexamĠwithĠ\(ĠkĠ>Ġ1Ġ\
Rollout 93, Act: 3.064
{2^{2014}}}ĊĊ<|im_start|>answerĊToĠcomputeĠtheĠsumĠ\(\sum_{k
Rollout 92, Act: 3.058
.78}Ġdiopters.ĊĊ<|im_start|>answerĊTheĠstudentĠisĠfarsightedĠandĠusesĠglassesĠto
Rollout 32, Act: 3.038
\boxed{4018}ĊĊ<|im_start|>answerĊGivenĠtheĠmatrixĠ\(\mathbf{M}
Rollout 84, Act: 3.030
ĠAnswer**Ċ\boxed{37}ĊĊ<|im_start|>answerĊLetĠtheĠinitialĠamountĠGaryĠpurchasedĠbeĠ1Ġunit
Rollout 23, Act: 2.998
FinalĠAnswer**Ċ\boxed{B}ĊĊ<|im_start|>answerĊToĠdetermineĠtheĠactivityĠofĠtheĠalphaĠdecayĠafterĠ
Rollout 70, Act: 2.992
**Ċ\boxed{315}ĊĊ<|im_start|>answerĊAĠpositiveĠintegerĠ\(ĠNĠ\)ĠisĠcalledĠa
Rollout 35, Act: 2.971
**Ċ\boxed{418}ĊĊ<|im_start|>answerĊGivenĠaĠpolynomialĠ\(ĠP(x)Ġ\)Ġwith
Rollout 46, Act: 2.969
Ġ\frac{dx}{dt}}Ċ$$ĊĊ<|im_start|>answerĊToĠexpressĠ\(\frac{d^{2
Rollout 25, Act: 2.969
}}$ĠtoĠmakeĠcontactĠwithĠEarth.ĊĊ<|im_start|>answerĊToĠdetermineĠhowĠmassiveĠtheĠasteroidĠ202
Rollout 9, Act: 2.965
ĠAnswer**Ċ\boxed{25}ĊĊ<|im_start|>answerĊSquareĠ\(AIME\)ĠhasĠsidesĠofĠlengthĠ
Rollout 3, Act: 2.961
{169}Ġm/s.ĊĊ<|im_start|>answerĊToĠcomputeĠtheĠmeanĠmolecularĠspeedĠ\(ĠvĠ\)
Rollout 80, Act: 2.942
ĠAnswer**Ċ\boxed{11}ĊĊ<|im_start|>answerĊToĠdetermineĠtheĠtimeĠscaleĠ\(ĠtĠ\)Ġit
Rollout 41, Act: 2.936
Ġ\boxed{927}.ĊĊ<|im_start|>answerĊToĠdetermineĠhowĠmanyĠpositiveĠintegersĠlessĠthanĠ1
Rollout 64, Act: 2.923
9}Ġ+Ġ1\right)}ĊĊ<|im_start|>answerĊGivenĠaĠsequenceĠofĠpositiveĠintegersĠ\(a_1
Top 16 Negative Activations
Rollout 22, Act: -4.040
ĠspeedĠisĠsqrt(8kT/(ÏĢm)).ĠSoĠtheĠuserĠisĠaskingĠforĠtheĠmeanĠspeed,
Rollout 3, Act: -4.057
^2Ġ=Ġ28,561.ĠSoĠsqrt(28,436)
Rollout 80, Act: -4.078
0^1ĠisĠ10^25.ĠSoĠifĠIĠhaveĠ92.64
Rollout 59, Act: -4.095
Ġ1/7Ġ=Ġ4/28.ĊĊLetĠmeĠverifyĠthatĠtheseĠsumĠupĠtoĠ1
Rollout 61, Act: -4.109
3ĊĊ133:7*19ĊĊNoĠcommonĠfactors,ĠsoĠgcd(7*1
Rollout 2, Act: -4.111
)^{997}Ġmod125ĊĊBecauseĠfromĠk=3ĠtoĠk=99
Rollout 0, Act: -4.139
,17,19ĠhaveĠexponentĠ1.ĊĊSoĠprimesĠ2:Ġ18,Ġ
Rollout 80, Act: -4.145
Ġ10^25Ġg.ĠGotĠit.ĠLetĠmeĠconfirm:ĊĊ12.8Ġ*Ġ
Rollout 2, Act: -4.157
kĠ-1)Ġâī¡Ġ-1Ġmod8.ĊĊTherefore,ĠforĠk=1:Ġ(2^
Rollout 17, Act: -4.192
48ÃĹ10^-33Ġm3.ĊĊTherefore,ĠtheĠentireĠ|I|=Ġ(11
Rollout 31, Act: -4.198
Ġquadratic:Ġx=(1Ġ±âĪļ5)/2.ĊĊThus,ĠtheĠrootsĠofĠtheĠcubicĠareĠx=-
Rollout 51, Act: -4.208
ĠwhenĠθĠisĠÏĢ.ĠSoĠtheĠlimitsĠwork.ĠTherefore,ĠIĠcanĠsplitĠtheĠintegralĠintoĠthreeĠparts
Rollout 2, Act: -4.212
ĠPĠâī¡109Ġmod125.ĊĊSoĠsummaryĠsoĠfar:ĊĊPâī¡5Ġmod8
Rollout 89, Act: -4.220
3/5Ġ*ĠTĠ=Ġ100.ĊĊTherefore,ĠTĠ=Ġ(100Ġ*5
Rollout 94, Act: -4.251
2i)*(-2)Ġ=Ġ-4iĊĊYes,Ġ-4iĠequalsĠ-4i.
Rollout 22, Act: -4.405
_meanĠ=Ġsqrt(8kT/(ÏĢm))ĊĊAlright,ĠsoĠfirstĠIĠneedĠtoĠfindĠm,

UP_PROJ

Top 16 Positive Activations
Rollout 84, Act: 2.412
32Ġ=Ġ37Ġ\).ĊĊTheĠfinalĠanswerĠisĠ\(\boxed{37}\).<|im_end|>
Rollout 69, Act: 2.243
2Ġ=Ġ204\).ĊĊTheĠfinalĠanswerĠis:Ċ\[Ċ\boxed{204
Rollout 73, Act: 2.120
398Ċ\]ĊĊThus,ĠtheĠfinalĠanswerĠis:Ċ\[Ċ\boxed{398
Rollout 47, Act: 2.118
Ġ=Ġ9ĊĠĠĠ\]ĊĊThus,ĠtheĠvalueĠofĠ\(ĠNĠ\)ĠisĠ\(\boxed{
Rollout 71, Act: 2.087
Ġ98Ċ\]ĊĊThus,ĠtheĠfinalĠanswerĠis:Ċ\[Ċ\boxed{98}Ċ
Rollout 83, Act: 2.069
Ġ144Ċ\]ĊĊThus,ĠtheĠnumberĠofĠdistinguishableĠacceptableĠarrangementsĠisĠ\(\boxed{
Rollout 88, Act: 2.044
Ġ100\).ĊĊThus,ĠtheĠtotalĠnumberĠofĠweightsĠdefinedĠforĠtheĠentireĠactivationĠoutputĠofĠthe
Rollout 95, Act: 2.027
490}\)Ġrational.ĊĊThus,ĠtheĠnumberĠofĠsegmentsĠ\(\overline{P_kQ
Rollout 32, Act: 2.025
Ġ4Ġ=Ġ4018Ġ\]ĊĊTheĠsumĠofĠtheĠentriesĠofĠ\(\mathbf{
Rollout 81, Act: 2.020
frac{510}{11}\).ĊĊTheĠareaĠofĠtriangleĠ\(ABC\)ĠisĠ\(\boxed
Rollout 84, Act: 2.020
Ġ4Ġ-Ġ4xĊ\]ĊĊ\[Ċ9Ġ-Ġ4Ġ=Ġ36xĠ-
Rollout 36, Act: 2.018
192Ċ\]ĊĊThus,ĠtheĠfinalĠanswerĠisĠ\(\boxed{192}\).
Rollout 32, Act: 2.002
4Ġ=Ġ4018Ġ\]ĊĊTheĠsumĠofĠtheĠentriesĠofĠ\(\mathbf{M
Rollout 4, Act: 1.973
0Ġ+Ġ3000}Ġ=Ġ\frac{6000000}{5
Rollout 73, Act: 1.966
00Ġ=Ġ398Ċ\]ĊĊThus,ĠtheĠfinalĠanswerĠis:Ċ\[Ċ\boxed
Rollout 60, Act: 1.961
,Ġ\text{Hz}Ċ\]ĊĊTheĠbeatĠfrequencyĠisĠtheĠdifferenceĠbetweenĠtheĠreflectedĠfrequencyĠ\(Ġf''
Top 16 Negative Activations
Rollout 79, Act: -3.002
ĠfromĠtheseĠlettersĠinclude:ĊĊ-Ġ"Hernias"ĠplusĠsomeĠothers.ĠNo.ĠMaybeĠ"Chairmen
Rollout 49, Act: -3.003
Ġsets)ĠisĠcalledĠaĠ"perfectlyĠnormal"Ġspace?ĠNotĠexactly.ĠPerfectlyĠnormalĠmeansĠthatĠevery
Rollout 11, Act: -3.003
ĠnumberĠisĠreal,ĠitsĠcomplexĠconjugateĠisĠitself.ĠSoĠforĠpĠtoĠbeĠreal,ĠpĠ=Ġconjug
Rollout 0, Act: -3.018
,Ġ3^8,Ġ5^4,Ġetc.,ĠupĠtoĠtheĠprimesĠlessĠthanĠorĠequal
Rollout 12, Act: -3.030
2?ĊĊAlternatively,ĠsubstitutionĠÏĨĠwhereĠtĠ=Ġcosθ?ĠLetĠmeĠthink.ĠForĠifĠtĠisĠfrom
Rollout 81, Act: -3.034
aticĠformula:ĠhĠ=Ġ[Ġ-70Ġ±Ġsqrt(70^2Ġ-Ġ4*1
Rollout 62, Act: -3.043
'sĠfamousĠpolynomialĠn^2Ġ+nĠ+41,ĠwhichĠgeneratesĠprimesĠforĠn=0ĠtoĠ
Rollout 54, Act: -3.053
ĠstartĠwithĠethanethiolateĠ(mostĠreactive)Ġ>ĠhydroxideĠ>Ġpropionate?ĠNo,
Rollout 93, Act: -3.071
14}/sqrt(pi*1007)}Ġ}ĠbyĠtheĠapproximationĠofĠcentralĠbinomialĠcoefficient:
Rollout 12, Act: -3.074
Ġdiverging.ĊĊAlternatively,ĠwithĠsubstitutionĠtĠ=Ġsinθ,ĠbutĠseemsĠnotĠapplicableĠhere.ĠWait,Ġlet
Rollout 91, Act: -3.094
2Ġ-ĠAC2)/(4Ġ*ĠareaĠofĠtriangle).ĠWait,ĠnotĠexactly.ĠLet'sĠrecallĠthatĠin
Rollout 62, Act: -3.123
Ġ-mĠ+11.ĠFromĠm=1ĠtoĠ10,Ġprimes.ĠLetĠmeĠcheckĠm
Rollout 68, Act: -3.153
's:Ġk=8Ġ(valid)ĠandĠkĠinĠk>8ĠandĠk<0Ġ(butĠneed
Rollout 35, Act: -3.154
ĠsolutionsĠtoĠthisĠquadraticĠareĠn=(41Ġ±Ġsqrt(1681Ġ-1672
Rollout 98, Act: -3.202
Ġ0.9ĠsolarĠmassesĠwouldĠbeĠaĠG-typeĠorĠmaybeĠK-type.ĠSoĠaĠ0.
Rollout 34, Act: -3.279
ĠbothĠequationsĠareĠVolterraĠintegralĠequationsĠofĠtheĠsecondĠkind,Ġright?ĠTheĠkernelĠisĠ\(Ġe^{-2

DOWN_PROJ

Top 16 Positive Activations
Rollout 96, Act: 4.978
06*169=ĠLet'sĠcompute:ĊĊ4106*100=4
Rollout 4, Act: 4.800
ĠLetĠmeĠcalculateĠ4.327ĠdividedĠbyĠ1200.ĊĊDivideĠbyĠ
Rollout 4, Act: 4.714
ĠcomputeĠthatĠmoreĠaccurately.ĊĊ4.327Ġ/Ġ1200.ĠLetĠmeĠcalculateĠ
Rollout 96, Act: 4.388
Ġ:ĊĊ-4106*169=ĠLet'sĠcompute:ĊĊ4106*1
Rollout 81, Act: 4.378
;ĠtoĠcomputeĠ44*561:ĊĊFirst,ĠthinkĠ561Ġ*Ġ4
Rollout 22, Act: 4.311
8.7768Ġ÷Ġ0.006333ĠâīĪĠ1
Rollout 36, Act: 4.264
,624*30,624:ĊĊ30,000*30
Rollout 22, Act: 4.193
2Ġ*Ġ273Ġ=Ġlet'sĠcompute:ĊĊ66.512Ġ*Ġ20
Rollout 3, Act: 4.175
=66.48*298=ĠsameĠapproach:Ġ66.48*
Rollout 92, Act: 4.137
37*123.ĠLetĠmeĠcompute:ĊĊ37*120Ġ=Ġ44
Rollout 36, Act: 4.135
584):ĊĊ147456Ġ-Ġ55584Ġ=ĠLet'sĠcompute
Rollout 3, Act: 4.090
6.512Ġ*Ġ298:ĊĊLetĠmeĠcomputeĠthisĠmultiplication.ĊĊ66.5
Rollout 80, Act: 4.084
.8Ġ*Ġ7.238ĠisĠ92.6464,Ġyeah,
Rollout 4, Act: 3.968
0Ġ/Ġ5025.ĠĊĊDividingĠ6,060,150
Rollout 22, Act: 3.957
6.512Ġ*Ġ273Ġ=Ġlet'sĠcompute:ĊĊ66.512
Rollout 17, Act: 3.951
42/1395.18âīĪ0.02538.ĠYes.ĊĊ
Top 16 Negative Activations
Rollout 56, Act: -3.063
ĠareaĠS_n=K_n=Ġ(sumĠfromĠk=1ĠtoĠn+1ĠofĠ1/
Rollout 89, Act: -3.073
333...ĠTherefore,Ġ2ĠpartsĠgirls:Ġ66.666...ĊĊAgainĠsame
Rollout 82, Act: -3.095
ĠcodewordĠisĠlength2,ĠsumĠis2*(allĠprobs)=2*69/69
Rollout 24, Act: -3.130
C^{\prime}Ġ\subseteqĠC$,ĠweĠhaveĠ$R^{\prime}=R$ĠandĠ$
Rollout 56, Act: -3.142
).ĊĊDistanceĠformula:ĊĊA3A1Ġ=Ġsqrt[(1Ġ-Ġ0)^2Ġ+Ġ(0Ġ-
Rollout 73, Act: -3.162
ĠwithĠn=1:ĠsumĠshouldĠbeĠ1*2Ġ=2.ĊĊPlugĠinĠn=1:
Rollout 18, Act: -3.168
+5k))/5Ġ=Ġ(-3Ġ-12Ġ-20k)/5Ġ=Ġ(-15
Rollout 67, Act: -3.169
ĠThenĠgivenĠthat:ĊĊForĠn=1:ĠA(1)^2Ġ+ĠB(1)Ġ+ĠC
Rollout 24, Act: -3.212
subseteqĠC$,ĠweĠhaveĠ$R^{\prime}=R$ĠandĠ$C^{\prime}=C
Rollout 45, Act: -3.248
Similarly,Ġf(2)=20:ĊĊk(8)Ġ+m(4)Ġ+n(
Rollout 93, Act: -3.249
ĠtheĠsumĠisĠ(C(2014,0)*1007)/2^{2
Rollout 45, Act: -3.293
Ġf(1)=1,ĠwhichĠgives:ĊĊk(1)^3Ġ+m(1)^2Ġ+
Rollout 44, Act: -3.354
2Ġ-ĠV1)Ġ=Ġ2*(2Ġ-1)=2ĠL·atm.ĠSoĠindeed
Rollout 24, Act: -3.470
Ġ\subseteqĠC$,ĠweĠhaveĠ$R^{\prime}=R$ĠandĠ$C^{\prime}=
Rollout 14, Act: -3.542
,Ġr=4,ĠsinceĠ1/2Ġ+1/4Ġ+1/4=1.ĊĊ
Rollout 95, Act: -3.614
1})Ġ=Ġ(kĠ-Ġ(k-1))*(AreaĠABC)/2450Ġ=ĠAreaĠABC

Layer 47

GATE_PROJ

Top 16 Positive Activations
Rollout 68, Act: 3.173
)$ĠhasĠexactlyĠoneĠrealĠsolution.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠthe
Rollout 97, Act: 3.140
ĠofĠ$\mathrm{Y}$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠhave
Rollout 47, Act: 3.138
ĠtheĠvalueĠofĠ$N$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠthere'sĠthisĠproblemĠabout
Rollout 91, Act: 3.130
Ġ-Ġ\cotĠC|.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠgeometryĠproblem
Rollout 66, Act: 3.117
$Ġ.ĠFindĠ$p$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthrough
Rollout 86, Act: 3.112
}}f(x)\Ġdx.$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠtheĠproblemĠisĠaskingĠfor
Rollout 24, Act: 3.107
ĠcontainĠtheĠsameĠnumberĠofĠrows.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠstartĠbyĠtryingĠto
Rollout 36, Act: 3.092
}$Ġ.ĠFindĠ$AP.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠout
Rollout 40, Act: 3.061
p|Ġ+Ġ|q|$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblemĠstep
Rollout 30, Act: 3.054
+3}}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'veĠgotĠthisĠproblem
Rollout 17, Act: 3.045
}|$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackleĠthis
Rollout 61, Act: 3.042
ĠFindĠ$p+q.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠsee.ĠSoĠthe
Rollout 1, Act: 3.042
Ġ\mathcal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthat
Rollout 31, Act: 3.032
ĠpossibleĠvaluesĠofĠ$y$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠtackleĠthis
Rollout 32, Act: 3.027
2009}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠtoĠfindĠthe
Rollout 72, Act: 3.026
ĠĠ$ĠĊ1.Ġ$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠtackleĠthisĠproblem:
Top 16 Negative Activations
Rollout 89, Act: -4.050
00ĠmultipliedĠbyĠ5ĠisĠ500,ĠdividedĠbyĠ3ĠisĠapproximatelyĠ16
Rollout 20, Act: -4.055
6ĠdividesĠ966?Ġ46ĠdividesĠintoĠ966?Ġ46Ġ*
Rollout 8, Act: -4.058
Ġ÷240=Ġ126.ĠSoĠ31/126.ĊĊWait
Rollout 3, Act: -4.060
).ĠLet'sĠsee,Ġ168^2Ġ=Ġ28,224.Ġ
Rollout 79, Act: -4.064
Ġ8-letterĠwordĠfollowedĠbyĠaĠ4-letterĠword.ĠLet'sĠtryĠtoĠfindĠpossibleĠcombinations.ĠMaybe
Rollout 59, Act: -4.071
ĠdividedĠbyĠ784.ĠReduction:ĠGCD(246,784).ĊĊPrime
Rollout 76, Act: -4.073
17ĠandĠ512ĠareĠcoprime.ĊĊSinceĠ512ĠisĠ2^
Rollout 79, Act: -4.085
ĠtoĠgetĠtheĠanswer.ĠTherefore,ĠsolvingĠanĠanagramĠofĠ"minorchangesin"Ġ(consideringĠ"
Rollout 19, Act: -4.115
ĠcubesĠupĠto2:1+8=9.Ġ9ĠdividedĠby7ĠgivesĠremainder9-7
Rollout 3, Act: -4.131
,820.576.ĊĊDivideĠthatĠbyĠÏĢĠ*Ġ0.222
Rollout 8, Act: -4.141
126.ĠThenĠcheckĠifĠtheyĠcanĠbeĠsimplified.ĠSinceĠ31ĠisĠprime.Ġ1
Rollout 19, Act: -4.152
Ġcubes1+8+27=36.ĠDividedĠbyn+5=8,Ġremainder
Rollout 81, Act: -4.241
Ġ=Ġlet'sĠcheck:Ġ172^2=29584ĠbecauseĠ17
Rollout 16, Act: -4.276
0ĠandĠ1953ĠareĠcoprime.ĠAsĠpreviousĠcalculationĠusingĠgcdĠstepsĠshowsĠthatĠgcd
Rollout 76, Act: -4.300
2.ĠButĠ417Ġ+512ĠwouldĠbeĠ929.ĊĊHence,
Rollout 59, Act: -4.340
2.ĠThenĠ123Ġ+392=Ġ515.ĊĊButĠlet'sĠcheck

UP_PROJ

Top 16 Positive Activations
Rollout 66, Act: 4.459
ĠisĠodd,ĠthenĠmĠwouldĠbeĠoddĠ+Ġeven?ĠWait,Ġm=12n2Ġ+Ġn
Rollout 70, Act: 4.293
331Ġ+16=331?ĠWait,ĠwaitĠN=35*9Ġ+
Rollout 32, Act: 4.264
Ġ+0Ġ+Ġ(âĪĴÏī2ĠfromĠS)Ġ??ĠWaitĠno.ĠLetĠmeĠrecomputetheĠpreviousĠstep
Rollout 93, Act: 4.256
=1:Ġ3/2,Ġnot.ĠWait,ĠforĠm=1,ĠsumĠisĠ(1
Rollout 96, Act: 4.208
7+6+9+3=34?ĊĊWait,Ġwait:ĊĊ5+8=13
Rollout 71, Act: 4.180
8.ĠIsĠ2008ĠaĠsquare?Ġsqrt(2008)=~44
Rollout 33, Act: 4.171
.ĊĊThus,Ġtotal:3+2=3?ĠWait,ĠtheĠentireĠpermutations:ĊĊTotalĠn=3
Rollout 91, Act: 4.169
2t2Ġ-2tĠ+1}/2Ġ?ĊĊWait,ĠagainĠsimilarĠsteps.ĠLet'sĠgoĠthrough
Rollout 96, Act: 4.161
Ġ+1134)/169.ĊĊWait,ĠinĠpreviousĠcalculations,ĠOB_xĠwasĠ(-8
Rollout 95, Act: 4.151
750=2*3675?ĊĊĠ3675*2=73
Rollout 22, Act: 4.136
732.4ĠatĠnumerator,ĠisĠthat?ĠNo,ĠnumeratorĠisĠ18,15
Rollout 48, Act: 4.119
Ġsum,Ġit'sĠFĠBĠ1Ġ2?ĠWait,Ġlet'sĠcomputeĠeachĠdigit:ĊĊRightmost
Rollout 39, Act: 4.091
10^3)/9Ġ-Ġ(3).ĠWait,ĠifĠweĠfollowĠtheĠformulaĠasĠbefore,Ġsum
Rollout 20, Act: 4.083
7,Ġ46ĠdividesĠ966?Ġ46ĠdividesĠintoĠ966?
Rollout 0, Act: 4.053
4,7:2,ĠwhichĠareĠallĠeven?ĠWait,Ġ18,8,4,
Rollout 66, Act: 4.050
Ġodd,ĠthenĠmĠwouldĠbeĠoddĠ+Ġeven?ĠWait,Ġm=12n2Ġ+Ġn.
Top 16 Negative Activations
Rollout 26, Act: -2.330
answerĊToĠfindĠtheĠstandardĠmolarĠenthalpyĠofĠformationĠofĠ$\mathrm{HgO}(
Rollout 57, Act: -2.341
thinkĊOkay,ĠsoĠIĠneedĠtoĠshowĠthatĠifĠweĠdefineĠaĠneighborhoodĠbasisĠaroundĠaĠpointĠxĠon
Rollout 15, Act: -2.353
ĠtheĠbehaviorĠofĠtheĠsystemĠusingĠlinearĠalgebraĠoverĠtheĠfieldĠ\(Ġ\text{GF}(2)Ġ\).
Rollout 77, Act: -2.358
}{4}Ġ\cdotĠf(10,Ġ4)Ġ\)Ċ-Ġ\(Ġf(10
Rollout 87, Act: -2.360
boxed{Ġ\dfrac{3Ġ\rhoĠa^2}{4Ġ\piĠ\epsilon_0}
Rollout 46, Act: -2.362
dt}}Ċ\]ĊandĊ\[Ċ\boxed{\frac{d^{2}Ġy}{dĠt
Rollout 10, Act: -2.384
ĠĠĠ-ĠAĠpositionĠisĠlosingĠifĠallĠpossibleĠmovesĠleadĠtoĠwinningĠpositions.ĊĠĠĠ-ĠThroughĠanalysis,Ġwe
Rollout 46, Act: -2.396
}{\partialĠy}Ġ-Ġ2Ġ\omegaĠ\frac{dx}{dt}}Ċ\]<|im_end|>
Rollout 74, Act: -2.414
Ġ\thetaĠ=ĠK_bĠ\sinĠ\thetaĠ+ĠK_sĠ\cosĠ\theta}Ġ\]<|im_end|>
Rollout 70, Act: -2.415
ĠwhenĠreadĠasĠaĠbase-10Ġnumber,ĠequalĠtwiceĠ\(ĠNĠ\).ĠToĠfindĠtheĠlargestĠsuch
Rollout 82, Act: -2.422
0Ċ-Ġ\(Ġ\frac{12}{69}Ġ\):Ġ90Ċ-Ġ\(
Rollout 67, Act: -2.462
2Ġ+Ġ492(8)Ġ+Ġ1020Ċ\]Ċ\[Ċc
Rollout 93, Act: -2.486
1ĠtoĠ1007ĠofĠcosineĠraisedĠtoĠtheĠ2014thĠpowerĠofĠ(
Rollout 95, Act: -2.524
ĠtheĠtriangleĠintoĠ2450ĠregionsĠofĠequalĠarea.ĠWeĠneedĠtoĠfindĠtheĠnumberĠofĠsegments
Rollout 0, Act: -2.544
}\)ĠandĠcalculateĠtheĠproductĠofĠtheĠresultingĠnumeratorĠandĠdenominator.ĠWeĠneedĠtoĠdetermineĠhowĠmanyĠsuchĠrational
Rollout 11, Act: -2.564
{b}{c}Ġ+Ġ\frac{c}{a}\),ĠgivenĠthatĠbothĠ\(p\)Ġand

DOWN_PROJ

Top 16 Positive Activations
Rollout 41, Act: 3.574
Ġa,Ġthree-digitĠnumbers.ĠSoĠ3ĠperĠa,ĠsoĠ9*3=27.ĊĊ
Rollout 56, Act: 3.321
ĠthisĠimpliesĠthatĠeachĠnextĠtermĠisĠmultipliedĠbyĠ(something)ĠfollowingĠtheĠformula.ĊĊButĠthisĠpatternĠisĠnot
Rollout 80, Act: 3.271
ĠSpecificĠheat:Ġ0.4ĠJ/gK.ĠDeltaĠT=1.ĠSoĠQ=9.
Rollout 4, Act: 3.224
0.52Ġ+0.52Ġ+something)%?ĠWait,ĠnoâĢĶwait:ĊĊWait,
Rollout 80, Act: 3.212
.265e25Ġ*0.4Ġ=3.706e25ĠJ
Rollout 81, Act: 3.189
.ĠWait,ĠleadingĠcoefficientsĠdiffer.ĠHuh.ĠWhat'sĠcausingĠthis?ĊĊWait,ĠinĠtheĠcoordinateĠsystem
Rollout 41, Act: 3.168
,Ġthree-digitĠnumbers.ĠSoĠ3ĠperĠa,ĠsoĠ9*3=27.ĊĊTherefore
Rollout 71, Act: 3.165
Now,ĠlookingĠforĠcĠsuchĠthatĠ3c2Ġ+Ġk2Ġ=Ġ8044.
Rollout 45, Act: 3.153
:ĠcĠ=Ġ0ĊĊSoĠcĠ=Ġ0.ĠThenĠfromĠEquationĠ1:Ġ2a
Rollout 70, Act: 3.129
2*2400=4800âīł6666,ĠsoĠthatĠdoesn't
Rollout 70, Act: 3.125
ĠbutĠ2*342=684âīł666.ĠSoĠno.ĊĊOr6
Rollout 13, Act: 3.101
^2.ĊĊPerhapsĠsetĠaĠ=Ġ1.25,Ġb=0.0044
Rollout 65, Act: 3.099
.ĠThereforeĠxĠ/ĠaĠ=Ġ1,ĠsoĠxĠ=Ġa,ĠwhichĠdoesn'tĠmakeĠsense.ĠAlternatively
Rollout 51, Act: 3.091
ĠTherefore,ĠperhapsĠweĠcanĠshiftĠtheĠvariableĠyĠ=ĠxĠ-Ġθ,ĠsoĠdyĠ=Ġdx,Ġand
Rollout 70, Act: 3.049
9,ĠwhichĠisĠlessĠthanĠ343?ĠWaitĠ499ĠisĠhigherĠthanĠ3
Rollout 85, Act: 3.021
ĠtheĠdirectionĠsteps:ĊĊIfĠf(x,y+1)Ġ-Ġf(x,y)Ġ=Ġ1ĠandĠf
Top 16 Negative Activations
Rollout 83, Act: -3.198
ĠAlso,ĠarrangementsĠthatĠcanĠbeĠrotatedĠintoĠeachĠotherĠareĠconsideredĠtheĠsame,ĠsoĠweĠhaveĠtoĠaccountĠfor
Rollout 13, Act: -3.213
1.12Ċx1.12Ċ--------Ċ224Ġ(1.12Ġ*
Rollout 99, Act: -3.224
3.199Ġ+3âīĪ0.505Ġ-3.199Ġ+3
Rollout 22, Act: -3.226
ĠthatĠtheyĠdidn'tĠsayĠtheĠtemperature,ĠthisĠproblemĠisĠincomplete.ĊĊWait,ĠbutĠperhapsĠinĠtheĠprocessĠof
Rollout 17, Act: -3.256
·m2/s2)Ġ)Ġ/ĠT?ĠWait,ĠmaybeĠit'sĠsimplerĠtoĠnoteĠthatĠunitsĠcheck
Rollout 31, Act: -3.270
1+\sqrt{5}}{2}}\),ĠandĠ\(\boxed{\frac{1-\sqrt
Rollout 73, Act: -3.282
=1,Ġ4-1=3,Ġ4-2=2.ĠSumĠisĠ1
Rollout 99, Act: -3.317
)Ġ=Ġf(0.5)ĠâīĪĠ2.19897ĊĊf(x
Rollout 42, Act: -3.326
coholsĠareĠtreatedĠwithĠacid.ĠPhosphoricĠacidĠisĠaĠstrongĠacid,ĠandĠinĠsuchĠconditions,
Rollout 0, Act: -3.344
2,3,5:ĊĊTheĠ8Ġassignments:ĊĊ1.ĠAssignĠnoneĠtoĠaĠ=>Ġa=1
Rollout 62, Act: -3.511
ĠthatĠhasĠatĠleastĠfourĠprimeĠfactors,Ġright?ĠTheyĠdon'tĠhaveĠtoĠbeĠdistinct,ĠsoĠevenĠif
Rollout 76, Act: -3.620
),Ġ(1,2),Ġ(2,1),ĠorĠ(2,2)ĠifĠweĠlabelĠthe
Rollout 99, Act: -3.811
1573)ĠâīĪĠ0.334Ġ(sinceĠ10^0.
Rollout 22, Act: -3.827
ĠofĠhydrogenĠgasĠ(H2)ĠatĠroomĠtemperature,ĠIĠguess?ĠTheĠquestionĠdoesn'tĠspecifyĠtheĠtemperature
Rollout 76, Act: -4.190
7ĠfactorsĠ3*139Ġ[sinceĠ1+4+1+7=13
Rollout 99, Act: -4.331
1575)ĠâīĪĠ0.334Ġ(sinceĠ10^0.

Layer 48

GATE_PROJ

Top 16 Positive Activations
Rollout 14, Act: 4.709
)Ġ\,ĠdxĠ\right)Ġ}ĊĊ<|im_start|>answerĊToĠproveĠtheĠinequalityĠĊĊ\[Ċ\left(\
Rollout 90, Act: 4.667
0891928}ĊĊ<|im_start|>answerĊToĠdetermineĠtheĠsmallestĠpositiveĠintegerĠ\(ĠAĠ\)
Rollout 9, Act: 4.624
ĠAnswer**Ċ\boxed{25}ĊĊ<|im_start|>answerĊSquareĠ\(AIME\)ĠhasĠsidesĠofĠlengthĠ
Rollout 81, Act: 4.591
{510}{11}}ĊĊ<|im_start|>answerĊGivenĠtriangleĠ\(ABC\)ĠwithĠ\(\tanĠ\
Rollout 95, Act: 4.572
FinalĠAnswer**Ċ\boxed{0}ĊĊ<|im_start|>answerĊTriangleĠ\(ABC\)ĠhasĠsideĠlengthsĠ\(ABĠ=
Rollout 91, Act: 4.556
FinalĠAnswer**Ċ\boxed{2}ĊĊ<|im_start|>answerĊGivenĠtriangleĠ\(ABC\)ĠwithĠmedianĠ\(AD\)
Rollout 0, Act: 4.533
**Ċ\boxed{128}ĊĊ<|im_start|>answerĊGivenĠaĠrationalĠnumberĠbetweenĠ0ĠandĠ1
Rollout 1, Act: 4.524
ĠyĠ\text{ĠexistsĠasĠdescribed}}ĊĊ<|im_start|>answerĊGivenĠanĠinfinite-dimensionalĠHilbertĠspaceĠ\(Ġ\math
Rollout 35, Act: 4.516
**Ċ\boxed{418}ĊĊ<|im_start|>answerĊGivenĠaĠpolynomialĠ\(ĠP(x)Ġ\)Ġwith
Rollout 16, Act: 4.503
{789953}ĊĊ<|im_start|>answerĊToĠdetermineĠtheĠexpectedĠnumberĠofĠdaysĠitĠwillĠtake
Rollout 2, Act: 4.468
**Ċ\boxed{109}ĊĊ<|im_start|>answerĊToĠfindĠtheĠremainderĠwhenĠ\(9Ġ\timesĠ
Rollout 31, Act: 4.465
{5}}{2}}\).ĊĊ<|im_start|>answerĊGivenĠrealĠnumbersĠ\(ĠxĠ\)ĠandĠ\(Ġy
Rollout 49, Act: 4.463
XĠ\text{ĠisĠcountable}}ĊĊ<|im_start|>answerĊToĠproveĠthatĠifĠeveryĠsubspaceĠofĠaĠHaus
Rollout 60, Act: 4.463
FinalĠAnswer**Ċ\boxed{6}ĊĊ<|im_start|>answerĊAĠstationaryĠsourceĠemitsĠsoundĠofĠfrequencyĠ\(Ġf_
Rollout 25, Act: 4.462
}}$ĠtoĠmakeĠcontactĠwithĠEarth.ĊĊ<|im_start|>answerĊToĠdetermineĠhowĠmassiveĠtheĠasteroidĠ202
Rollout 36, Act: 4.462
**Ċ\boxed{192}ĊĊ<|im_start|>answerĊGivenĠaĠcubeĠwithĠedgesĠ\(\overline{
Top 16 Negative Activations
Rollout 79, Act: -4.779
ĠchangesĠinĠinvestigators'Ġfacility"ĠisĠtheĠwordplay.ĠButĠtheĠwayĠit'sĠphrasedĠisĠ"
Rollout 87, Act: -4.782
ĠsetĠuĠ=Ġy,ĠvĠ=Ġz.ĠNotĠhelpful.ĠAlternatively,Ġnon-dimensionalizeĠtheĠvariablesĠbyĠdefining
Rollout 79, Act: -4.790
"ĠasĠ"knighted".ĠDoesn'tĠmakeĠsense.ĊĊAnotherĠapproach:ĠThinkĠofĠabbreviationsĠorĠhy
Rollout 48, Act: -4.795
xB612Ġ(wrongĠwayĠinĠpreviousĠstep).ĠWaitĠno,ĠtheĠthirdĠwordĠisĠ0xB
Rollout 11, Act: -4.812
Ġy/z,ĠsinceĠzĠisĠconjugateĠofĠc.ĠHmmm,ĠperhapsĠrequiringĠrelationshipsĠlikeĠa/bĠ=
Rollout 82, Act: -4.814
ĠThenĠpairĠtheĠnextĠtwo,Ġetc.ĠLet'sĠsee:ĊĊGroup1:Ġp1,Ġp5ĊĊ
Rollout 59, Act: -4.819
Ġ+4/28Ġz3ĊĊCombineĠlikeĠterms:ĊĊConstantĠterm:Ġ3/28ĊĊ
Rollout 93, Act: -4.888
=2*(1/2)Ġ+Ġ0?ĠNotĠsure.ĠWhenĠm=2:1Ġ(
Rollout 13, Act: -4.905
5.ĠAlternatively,ĠperhapsĠuseĠtheĠstandardĠmultiplicationĠgrid.ĠButĠthisĠmightĠtakeĠlonger.ĊĊAlternatively,ĠanotherĠmethod
Rollout 91, Act: -4.929
ĠcomputedĠbeforeĠmayĠbetter,ĠevenĠthoughĠmoreĠsteps.ĠLet'sĠproceed.ĊĊCalculatingĠangleĠatĠC:ĊĊVectors
Rollout 4, Act: -4.939
Wait,ĠthisĠisĠanotherĠwayĠofĠwritingĠtheĠterms.ĠMaybeĠthisĠwillĠincreaseĠprecision?ĊĊWait,ĠorĠrather
Rollout 73, Act: -4.970
,ĠthatĠseemsĠcircular.ĠMaybeĠthisĠmethodĠisn'tĠworking.ĊĊWait,ĠperhapsĠanotherĠapproach.ĠWait,Ġuse
Rollout 4, Act: -4.975
49e-3*sqrt(10).ĠBecauseĠsqrt(10)=3.16
Rollout 36, Act: -4.979
,000Ġ+Ġ(-864)ĊĊWait,ĊĊWait,ĠbetterĠtoĠuseĠequation(d):Ġs
Rollout 41, Act: -4.991
ĠthenĠsumĠthemĠupĠforĠallĠlengths.ĠLet'sĠsee.ĊĊDefineĠSĠasĠtheĠtotalĠacrossĠallĠlengthsĠ(
Rollout 11, Act: -5.111
ĠvariablesĠisĠtheĠproductĠofĠconjugatesĠofĠtheĠothers.ĠForĠinstance,ĠmaybeĠcĠ=Ġ\overline{

UP_PROJ

Top 16 Positive Activations
Rollout 63, Act: 3.595
,Ġ\(ĠpqĠ=Ġ0Ġ\),ĠsoĠtheĠbaseĠcaseĠisĠsatisfied.ĊĊTherefore,ĠitĠseemsĠthatĠ\(
Rollout 33, Act: 3.593
(n)ĠbeĠtheĠnumberĠofĠpermutationsĠofnĠelementsĠsatisfyingĠthatĠfor1Ġ<=kĠ<=n-1,
Rollout 73, Act: 3.588
00=398.ĊĊTherefore,ĠtheĠremainderĠshouldĠbeĠ398.ĊĊWait,Ġbut
Rollout 0, Act: 3.574
ĠaĠshareĠnoĠprimesĠwithĠb.ĠTherefore,ĠtheĠcoprimeĠpairsĠ(a,b)ĠwithĠa*b
Rollout 58, Act: 3.531
1Ġ*ĠP(0).ĠTherefore,ĠeachĠtermĠinĠtheĠsumĠforĠpartĠiĠisĠiĠ*Ġsomething...
Rollout 30, Act: 3.524
ĠspacedĠbyĠtwoĠindices.ĠSinceĠit'sĠanĠinfiniteĠsum,ĠmaybeĠitĠtelescopes?ĠIfĠthat'sĠthe
Rollout 41, Act: 3.519
ĠareĠallowed.ĠTherefore,ĠtotalĠtwo-digitĠnumbersĠthatĠmeetĠtheĠconditionsĠwouldĠbeĠallĠ90Ġnumbers,
Rollout 45, Act: 3.518
ĠbutĠf(x)ĠisĠcubic.ĠForĠpartĠ(b),ĠperhapsĠit'sĠonlyĠforĠnon-negativeĠintegersĠx
Rollout 72, Act: 3.504
Ġnotation.ĠIfĠunsure,ĠbutĠknowingĠtheĠproblemĠrequiresĠwritingĠallĠevenĠpositiveĠintegers,ĠtheĠsafestĠisĠtoĠstate
Rollout 15, Act: 3.498
Ġ4.ĠTherefore,ĠaccordingĠtoĠtheĠlinearĠalgebraĠmodel,ĠanyĠinitialĠstateĠshouldĠdieĠoutĠafterĠatĠmost
Rollout 68, Act: 3.492
ĠnoĠrealĠsolutions.ĊĊButĠweĠneedĠtoĠrememberĠthatĠtheseĠsolutionsĠmustĠalsoĠsatisfyĠxĠ>Ġ-2ĠandĠk
Rollout 76, Act: 3.481
95.ĠTherefore,ĠtotalĠcoloringsĠwithĠatĠleastĠoneĠ2x2ĠredĠsquareĠisĠ9
Rollout 89, Act: 3.474
Ġ66.666...ĊĊAgainĠsameĠanswer.ĊĊButĠagain,ĠthatĠgivesĠusĠaĠfractionalĠanswer
Rollout 89, Act: 3.470
'sĠprecise.ĠSoĠmaybeĠthisĠproblemĠisĠaĠtrickĠquestion,ĠbutĠtypicallyĠthough,ĠproblemsĠwouldĠhaveĠtheĠfractional
Rollout 13, Act: 3.468
ĠisĠ187.12ĠwhichĠseemsĠreasonable.ĊĊSoĠmyĠconclusionĠisĠthatĠtheĠextraĠamountĠJo
Rollout 16, Act: 3.467
ĠanswerĠisĠsupposedĠtoĠbeĠmĠandĠnĠcoprime,ĠsoĠm=7880ĠandĠn
Top 16 Negative Activations
Rollout 78, Act: -2.632
{v_eĠ-Ġ20}{60}Ġ\):ĊĠĠĠ\[ĊĠĠĠLĠ=Ġ\frac
Rollout 95, Act: -2.632
5=Ġsqrt(671*3)/5=Ġsqrt(2013)/5.ĊĊ
Rollout 64, Act: -2.693
Ġ(2^{2009}Ġ-1)Ġ+Ġa1(2Ġ-Ġ2^{2
Rollout 7, Act: -2.712
PĠ=79xĠ-Ġ(AĠ+5)xĊĊ-3P=(74Ġ-A)
Rollout 13, Act: -2.713
,ĠbĠ=Ġ0.1255ĊĊButĠthat'sĠnotĠhelpful.ĠLetĠmeĠcomputeĠ
Rollout 53, Act: -2.721
Ġ(Ġ3dĠ-Ġ2dĠ)Ġ/6Ġ=Ġd/6ĊĊTherefore,ĠPDĠdistanceĠis
Rollout 40, Act: -2.737
{2}Ġ+Ġ1))(1Ġ+Ġi)Ġ\]ĊExpandingĠtheĠproduct:Ċ\[Ġ(1
Rollout 95, Act: -2.754
3ĠsĠ-Ġ9t)Ġ-Ġ9t???ĠWait.ĊĊWait,ĠholdĠon.ĠLetĠme
Rollout 86, Act: -2.756
/2)(Ġ-Ġ(-1Ġ-1Ġ)Ġ/2Ġ)=Ġ-Ġ(1/2)(Ġ-Ġ(-
Rollout 87, Act: -2.764
(1Ġ+Ġr2)))Ġ*Ġ2rĠ]Ġ=ĠrĠ/Ġsqrt(1Ġ+Ġr2)ĊĊ
Rollout 78, Act: -2.802
frac{v_eĠ-Ġ20}{60}Ġ\):ĊĠĠĠ\[ĊĠĠĠLĠ=Ġ\
Rollout 81, Act: -2.853
70Ġ\pmĠ172}{22}Ġ\impliesĠhĠ=Ġ\frac{1
Rollout 11, Act: -2.862
ĠaĠbcĠ=1,ĠsoĠab*cĠ=a.ĠSoĠaĠbĠcĠ=1.ĠHence,Ġmultiplying
Rollout 71, Act: -2.944
Ġ(-aĠ-Ġb)aĠ+Ġ(-aĠ-Ġb)bĠ=ĠabĠ-Ġa^2Ġ-ĠabĠ-Ġab
Rollout 30, Act: -2.949
(n+3)-1}Ġ+1}Ġ)Ġ=Ġ?ĊĊWait,Ġno.ĠLet'sĠrecallĠthe
Rollout 7, Act: -3.121
=Ġ(56Ġ-54)N/3Ġ?ĊĊWait,Ġwait:ĊĊWaitĠ18N

DOWN_PROJ

Top 16 Positive Activations
Rollout 96, Act: 5.070
2ĠtoĠ132*(169/169)=Ġ22308
Rollout 61, Act: 4.463
ĠFindĠ$p+q.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,Ġlet'sĠsee.ĠSoĠthe
Rollout 89, Act: 4.422
ĠmanyĠgirlsĠareĠinĠtheĠclass?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠseeĠhere.ĠSo
Rollout 85, Act: 4.422
,2024)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'mĠtryingĠtoĠsolve
Rollout 16, Act: 4.419
/2)^6Ġ=Ġ1Ġ-Ġ(1/2)^6.ĠWait,Ġthat'sĠactuallyĠcorrect
Rollout 84, Act: 4.406
.ĠFindĠ$m+n$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠsolveĠthis
Rollout 20, Act: 4.400
ĠfactorizationĠofĠ$n$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠthe
Rollout 73, Act: 4.389
Ġ$1000$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠsolveĠthis
Rollout 35, Act: 4.379
\cdotĠn_2.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfindĠthe
Rollout 59, Act: 4.379
ĠFindĠ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠtheĠproblemĠisĠaboutĠJackie
Rollout 76, Act: 4.361
ĠFindĠ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠfindĠtheĠprobability
Rollout 2, Act: 4.358
Ġ$1000$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠproblemĠhere
Rollout 8, Act: 4.332
ĠFindĠ$mĠ+Ġn$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠWeĠneed
Rollout 30, Act: 4.325
+3}}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'veĠgotĠthisĠproblem
Rollout 55, Act: 4.323
2018}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠhaveĠthisĠmatrixĠA
Rollout 56, Act: 4.315
ĠanĠareaĠofĠ$K$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠWesyuĠisĠbuildingĠthis
Top 16 Negative Activations
Rollout 36, Act: -3.003
456Ġ-Ġ55584Ġ=ĠLet'sĠcomputeĠ147456
Rollout 21, Act: -3.006
Ġinteger.ĠForĠexample,Ġfloor(x)ĠisĠsurjectiveĠontoĠZ.ĠSimilarly,ĠcertainĠstepĠfunctions.ĠBut
Rollout 14, Act: -3.020
ĠdxĠ-ĠdĠ\intĠx^2ĠdxĠ=Ġc*(Ġ(Ġ(c/d)^2Ġ)Ġ/2
Rollout 37, Act: -3.032
ĠbutĠtheĠproblemĠstatesĠthatĠtheĠa_iĠareĠintegers,ĠbutĠnotĠnecessarilyĠdistinct,ĠandĠtheĠk_iĠare
Rollout 89, Act: -3.037
Ġboys?ĠButĠinĠrealĠlife,ĠyouĠcan'tĠhaveĠaĠfractionĠofĠaĠperson.ĠTherefore,ĠtheĠtotal
Rollout 14, Act: -3.041
Ġ$-\muĠK$ĠisĠ$-\muĠx$.ĠSoĠsettingĠtheĠfunctionalĠderivativeĠtoĠzeroĠgives:ĊĊ$$
Rollout 2, Act: -3.041
9:Ġ891*999=891*(1000Ġ-1
Rollout 52, Act: -3.063
ĠisĠcorrect.ĠWait,ĠtheĠproblemĠmentionsĠthatĠAĠhasĠtoĠbeĠpartĠofĠtheĠsequenceĠwhichĠaddsĠtoĠ
Rollout 21, Act: -3.092
)Ġ=ĠkĠforĠintegerĠk,ĠbutĠvariablesĠcanĠtakeĠanyĠrealĠnumber.ĠHowever,ĠfunctionalĠequationsĠthatĠare
Rollout 36, Act: -3.100
,362.666...,ĠwhichĠisĠnotĠaĠwholeĠnumber.ĠMaybeĠthereĠwasĠaĠcalculation
Rollout 2, Act: -3.210
9=Ġ891*999=Ġlet'sĠcompute:Ġ891*(1
Rollout 15, Act: -3.233
ĠwhetherĠthatĠsingleĠneighborĠisĠon.ĠForĠallĠotherĠlamps,ĠsinceĠtheyĠhaveĠtwoĠneighbors,ĠtheirĠnextĠstate
Rollout 37, Act: -3.238
ĠtheĠproblemĠstatesĠthatĠtheĠa_iĠareĠintegers,ĠbutĠnotĠnecessarilyĠdistinct,ĠandĠtheĠk_iĠareĠpositive
Rollout 67, Act: -3.503
Ġ2:Ġq=Ġ-72Ġ/Ġd.ĊĊTherefore,ĠsubstitutingĠintoĠequationĠ3:ĊĊa
Rollout 23, Act: -3.783
.073...ĠminâĪĴ1ĠâĢĵĠthat'sĠcorrect.ĠĊĊTherefore,Ġλ_totalĠâīĪĠ
Rollout 71, Act: -3.896
11).Ġsqrt(2011)ĠisĠapproxĠ44.86...ĊĊSoĠprimes

Layer 49

GATE_PROJ

Top 16 Positive Activations
Rollout 82, Act: 5.598
len2).ĊĊAndĠ15Ġis0,ĠwhichĠisĠlen1.ĊĊSoĠWait,ĠactualĠcodew
Rollout 6, Act: 5.365
ĠthenĠO(n^kĠ/ĠlogĠn),ĠbutĠthisĠisĠstillĠnotĠpolynomialĠperĠlength.ĠWait.ĊĊAlternatively
Rollout 75, Act: 5.351
ĠTherefore,ĠS=8NĠ-5.ĠButĠthisĠmayĠcomplicateĠthings.ĠWait,Ġno,Ġperhaps
Rollout 70, Act: 5.349
ĠsoĠ2*49=98Ġ(whichĠisĠtwo-digit)ĠtoĠ2*34
Rollout 38, Act: 5.166
,ĠwhichĠwouldĠmeanĠxĠ=2L,ĠbutĠthisĠwouldĠmakeĠpointĠD=(-2L,ĠW),
Rollout 6, Act: 5.148
ĠhaveĠp(n)ĠstringsĠforĠeachĠn,ĠbutĠthisĠmightĠnotĠbeĠsparseĠifĠp(n)ĠisĠa
Rollout 75, Act: 5.146
?ĠOnlyĠ1Ġ+Ġ8,ĠwhichĠ8ĠisĠnotĠaĠsquare;Ġ4Ġ+Ġ5
Rollout 15, Act: 5.059
ĠwhichĠisĠ2^2Ġ-1:Ġno,ĠsinceĠn=3ĠwasĠbad.ĠMaybeĠanotherĠpattern
Rollout 10, Act: 5.045
:Ġsubtract16=10Ġ(L)ĠwhichĠisĠprevious,ĠsoĠfromĠ26ĠcanĠgo
Rollout 28, Act: 5.039
ĠyĠ+a2b2=0ĊĊButĠthisĠgives:ĊĊ(2bĠxĠ+2aĠy
Rollout 75, Act: 5.037
6+16=32Ġalready.ĠButĠthat'sĠ16+16+0,
Rollout 49, Act: 5.012
-compact.ĠTherefore,ĠhereditaryĠÏĥ-compactnessĠisĠaĠstrongerĠcondition.ĊĊSoĠinĠourĠcase,
Rollout 11, Act: 5.008
0.ĠThenĠp=Ġ-2.ĠButĠmaybeĠthisĠisĠinvalid.ĊĊGivenĠpreviousĠproblemĠmaybeĠDavidĠAltizio
Rollout 28, Act: 4.980
ĊĊWhichĠisĠexactlyĠwhatĠweĠstartedĠwith.ĠSoĠthisĠtransformationĠdoesn'tĠhelp.ĠAlternatively,ĠequationĠofĠthe
Rollout 38, Act: 4.980
=2*(x/2)=x.ĠSoĠno,Ġit'sĠL=x/2.ĠButĠthen
Rollout 15, Act: 4.975
.ĠAllĠoff.ĠSoĠwithinĠoneĠstep!ĠButĠthisĠseemsĠcontradictory.ĠWait,ĠmaybeĠaĠmistake.ĊĊWait
Top 16 Negative Activations
Rollout 46, Act: -7.009
}+y^{2}\right)$.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠto
Rollout 25, Act: -7.010
Ġ$r$ĠisĠtheĠimpactĠparameter.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigure
Rollout 87, Act: -7.014
0}$,ĠandĠanyĠnecessaryĠnumericalĠconstants.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠto
Rollout 29, Act: -7.020
PĠ<Ġ\rhoĠgH$ĠĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigure
Rollout 57, Act: -7.021
.ĠčĊčĊ*A.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshow
Rollout 48, Act: -7.033
0Ġ.ĠDoesĠtheĠheaderĠcontainsĠerror?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠhaveĠthisĠIPv
Rollout 54, Act: -7.035
Ġ3,ĠBĠ=Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestion
Rollout 3, Act: -7.046
Ġ(Rn)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠtoĠcalculateĠthe
Rollout 22, Act: -7.060
Ġ(H2)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneedĠtoĠcalculate
Rollout 98, Act: -7.082
Ġ4Ċ(D)Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigure
Rollout 92, Act: -7.091
ĠrefractiveĠpowerĠofĠhisĠglassesĠlenses?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigure
Rollout 58, Act: -7.114
ProposedĠbyĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠSo
Rollout 23, Act: -7.154
7.555ĠBqĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠproblem
Rollout 94, Act: -7.175
Ġc,dĊ(D)Ġa,bĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠdetermine
Rollout 43, Act: -7.264
left|0\right\rangleĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigure
Rollout 42, Act: -7.302
4]oct-5-eneĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfigure

UP_PROJ

Top 16 Positive Activations
Rollout 95, Act: 4.122
ĠandĠQk.ĊĊSinceĠPkĠisĠonĠAB,ĠwhichĠgoesĠfromĠA(0,0)Ġto
Rollout 95, Act: 4.059
1/3).ĠSoĠweĠcanĠparametrizeĠPkĠandĠQkĠwithĠparameterĠt,ĠwhereĠtĠis
Rollout 72, Act: 3.753
Ġno.ĠTheĠquestionĠrequiresĠ"FindĠallĠnumbersĠn...",ĠhenceĠtheĠanswerĠisĠallĠevenĠintegersĠnĠâī¥2
Rollout 32, Act: 3.685
3,ĠandĠtheĠleftoverĠterms.ĊĊNumberĠofĠcycles:Ġ2009ĠdividedĠbyĠ3Ġgives
Rollout 95, Act: 3.628
0)ĠtoĠ(9,0),ĠsoĠAPkĠisĠ(9k/2450,
Rollout 95, Act: 3.616
ĠtĠisĠtheĠscalingĠfactor,ĠthenĠcoordinatesĠofĠPkĠwouldĠbeĠ(9t,Ġ0),Ġand
Rollout 95, Act: 3.607
ĠdividesĠABĠintoĠ2450ĠequalĠparts,ĠAPkĠ=Ġ(9/2450
Rollout 95, Act: 3.594
ĠrecastĠthis.ĊĊPointĠPkĠisĠalongĠAB.ĠABĠfromĠ(0,0)ĠtoĠ(9
Rollout 6, Act: 3.592
ĠSĠisĠsparse.ĊĊThen,ĠtheĠoracleĠmachineĠM,ĠwhenĠgivenĠ<x,i>,ĠencodesĠiĠand
Rollout 95, Act: 3.586
ĠisĠtheĠscalingĠfactor,ĠthenĠcoordinatesĠofĠPkĠwouldĠbeĠ(9t,Ġ0),ĠandĠcoordinates
Rollout 6, Act: 3.584
ĠbitĠofĠα_n.ĠThen,ĠtheĠmachineĠM,ĠgivenĠxĠofĠlengthĠn,ĠneedsĠtoĠgetĠthe
Rollout 61, Act: 3.581
ĠforĠd=9:ĠkĠ=ĠmĠ+9,ĠmĠcanĠgoĠfromĠ0ĠtoĠ39
Rollout 81, Act: 3.565
'sĠcomputeĠtheĠslopes.ĠĊĊTheĠslopeĠofĠAB:ĠsinceĠAĠisĠ(0,h)ĠandĠBĠis
Rollout 14, Act: 3.550
.ĊĊForĠI1=âĪ«0^AĠfĠdx:ĊĊĠapplyCauchy-Schwarz:ĠI
Rollout 95, Act: 3.547
Ġthings,ĠbutĠperhapsĠlet'sĠparametrizeĠpointsĠPkĠandĠQk.ĊĊSinceĠPkĠisĠonĠAB
Rollout 6, Act: 3.541
Ġsparse.ĊĊToĠretrieveĠα_n,ĠtheĠoracleĠmachine,ĠgivenĠinputĠxĠofĠlengthĠn,ĠwouldĠneedĠto
Top 16 Negative Activations
Rollout 70, Act: -1.899
2*343=1000Ġ-686=314.ĊĊi=
Rollout 36, Act: -1.912
728Ġ+Ġ37728Ġ+Ġ16128)s2Ġ=Ġ(
Rollout 70, Act: -1.926
,Ġ500Ġ<=NĠ<=4999.5.ĠButĠNĠisĠalsoĠaĠfour
Rollout 23, Act: -1.926
.88âīĪ0.6931/5.88âīĪ0.117
Rollout 96, Act: -1.932
169ĊĊ=Ġ(4563Ġ+1134)/169ĊĊ=
Rollout 5, Act: -1.940
Ġ3bĠ+Ġ75Ġ=Ġ2bĠ+Ġ150Ġ\impliesĠbĠ=
Rollout 96, Act: -1.944
)/169=Ġ(4563Ġ+1134)/169=5
Rollout 93, Act: -1.949
2Ġ/1007ĠâīĪĠ0.00987,ĠsoĠe^{-0
Rollout 4, Act: -1.953
}Ġ=Ġ\sqrt{12.96Ġ+Ġ5.76}Ġ=Ġ\sqrt{
Rollout 67, Act: -1.962
2Ġ+32*(-6)=540Ġ-192=348.ĠCorrect.ĊĊ
Rollout 16, Act: -1.967
864;Ġ64*31=1984.Ġ9864
Rollout 20, Act: -1.984
7,46)=47*46=2162.ĊĊTherefore,ĠnĠmustĠbe
Rollout 38, Act: -2.000
riminant=390,625ĠâĪĴ112,896=27
Rollout 39, Act: -2.004
9)/9Ġ=Ġ(10000Ġ-2899)/9Ġ=Ġ(71
Rollout 96, Act: -2.018
9.ĊĊSoĠx=(22308Ġ-13608)/169Ġ/
Rollout 93, Act: -2.034
1007ĠâīĪĠ0.00987,ĠsoĠe^{-0.0

DOWN_PROJ

Top 16 Positive Activations
Rollout 99, Act: 4.860
)2ĠâīĪĠ0.6389ĊĊ4ĠaĠcĠ=Ġ4*(-0.
Rollout 90, Act: 4.732
=14,126.ĠTotal=403,600Ġ+141
Rollout 4, Act: 4.674
3*25.2004=603*25Ġ+603*
Rollout 68, Act: 4.472
ĠmeĠthinkĠhowĠtoĠapproachĠthis.ĠAlright,Ġlogarithms...ĠtheĠequationĠinvolvesĠlogs,ĠsoĠfirstĠIĠshould
Rollout 76, Act: 4.462
Ġ2x2ĠentirelyĠredĠsquares.ĠItĠmightĠbeĠeasierĠtoĠcomputeĠtheĠnumberĠofĠcoloringsĠthatĠhave
Rollout 36, Act: 4.346
0Ġ+3,136=3,240,000+201
Rollout 62, Act: 4.194
Ġ11*97.ĊĊThus,ĠtheseĠhappenĠwhenĠmĠisĠ11kĠorĠ1
Rollout 99, Act: 4.193
5)2ĠâīĪĠ0.6389ĊĊ4ĠaĠcĠ=Ġ4*(-0
Rollout 4, Act: 4.181
004e-12ĠâīĪĠ603*25.2004
Rollout 80, Act: 4.166
inosĠpassĠthroughĠit?ĠSinceĠtheĠneutrinosĠareĠcomingĠfromĠtheĠSun,ĠtheĠfluxĠatĠEarth'sĠsurface
Rollout 35, Act: 4.154
ĠisĠanĠinteger.ĠButĠsinceĠaĠandĠbĠareĠseparatedĠbyĠ7,ĠweĠcanĠpossiblyĠenumerateĠallĠpairs
Rollout 90, Act: 4.154
Ġ1928ĠisĠaĠfour-digitĠnumber?Ġ1,928.ĠYes.ĠSo
Rollout 13, Act: 4.146
AddingĠthemĠtogether:Ġ1.0609Ġ+Ġ0.063654
Rollout 48, Act: 4.141
Ġnumbers,ĠwhenĠadding,ĠifĠthere'sĠaĠcarryĠbeyondĠ16Ġbits,ĠweĠneedĠtoĠaddĠthat
Rollout 87, Act: 4.139
.ĠLetĠmeĠsee.ĠIntegrationĠinĠthreeĠdimensionsĠcanĠsometimesĠbeĠsimplifiedĠbyĠusingĠsphericalĠcoordinates,ĠbutĠhereĠthe
Rollout 43, Act: 4.122
ĠisĠtheĠfirstĠqubitĠbeingĠ|0⣩.ĠThatĠmeansĠthisĠisĠaĠbitĠdifferentĠfromĠtheĠstandardĠC
Top 16 Negative Activations
Rollout 95, Act: -3.782
))Ġ=ĊĊ(Ġ(25/3Ġ-9)k/2450,Ġsqrt(6
Rollout 56, Act: -3.841
xĠ=Ġ0Ġ+Ġt*(1Ġ-Ġ0)Ġ=ĠtĊĊyĠ=Ġ(-âĪļ3/
Rollout 7, Act: -3.843
ĠthisĠmathĠproblemĠaboutĠparticipantsĠinĠaĠtest.ĠLetĠmeĠfirstĠtryĠtoĠunderstandĠwhat'sĠbeingĠaskedĠhere.ĊĊ
Rollout 86, Act: -3.867
1/2*(-1/2*(-1Ġ-1Ġ))=Ġ-1/2*(-1/2
Rollout 7, Act: -3.871
/3)Ġ=Ġ(75*2)/3Ġ*N=150/3Ġ*N
Rollout 93, Act: -3.928
2014}ĠC(2014,j)Ġe^{iθ(201
Rollout 56, Act: -3.934
Ġ=Ġ(1/(2^{2}Ġ-Ġ2))ĠA2A0ĠwhichĠisĠ1/(
Rollout 28, Act: -3.961
6(X2Ġ-Ġ(2âĪļ2Ġ/3)ĠXĠ)Ġ+2Y2Ġ+1=0
Rollout 45, Act: -3.975
)Ġ+Ġ1Ġ+ĠaĠx^2Ġ+ĠaĠxĠ+ĠbĠxĠ+ĠcĠxĠ+ĠcĠ+
Rollout 28, Act: -4.027
Ġ6(X2Ġ-Ġ(4âĪļ2Ġ/6)ĠXĠ)Ġ+2Y2Ġ+1=
Rollout 40, Act: -4.029
ĠcoefficientsĠĠ(-5Ġ+5(âĪļ2Ġ+2Ġ))iĠ=Ġ(5âĪļ2Ġ+5)i
Rollout 21, Act: -4.151
(x)Ġ=Ġcx,ĠsoĠc(cx)=cxĠimpliesĠc2xĠ=ĠcxĠforĠallĠx.ĠSo
Rollout 36, Act: -4.155
00/sĠ+Ġ7200/s)ĠxĠ=Ġ(25200/s)x
Rollout 28, Act: -4.364
6(X2Ġ-Ġ(4âĪļ2Ġ/6)ĠXĠ)Ġ+2Y2Ġ+1=0
Rollout 7, Act: -4.412
5)xĊĊ-3P=(74Ġ-A)Ġx.ĊĊWait,ĠsubstituteĠP=2N
Rollout 30, Act: -4.936
)Ġ-Ġ(b_{n+2}Ġ+1)Ġ=Ġa_{n+1}Ġb_{

Layer 50

GATE_PROJ

Top 16 Positive Activations
Rollout 82, Act: 2.673
0Ġ+16+16)/26âīĪ42/26âīĪ1.61
Rollout 41, Act: 2.208
\)ĊĠĠĠ-ĠTotalĠforĠ4-digitĠnumbers:Ġ\(9Ġ+Ġ504Ġ+Ġ6
Rollout 41, Act: 2.206
\)ĊĠĠĠ-ĠTotalĠforĠ3-digitĠnumbers:Ġ\(9Ġ+Ġ216Ġ+Ġ2
Rollout 84, Act: 2.206
Ġ36xĠ-Ġ4xĊ\]ĊĊ\[Ċ5Ġ=Ġ32xĊ\
Rollout 72, Act: 2.204
Ġ\),ĠandĠ\(Ġ-1Ġ\).ĠTheirĠsumĠisĠ\(Ġ1Ġ+Ġ1Ġ+Ġ(-1)
Rollout 72, Act: 2.187
Ġ\(Ġ1Ġ+Ġ1Ġ+Ġ(-1)Ġ=Ġ1Ġ\).ĊĠĠĠ-ĠThisĠconstructionĠworksĠfor
Rollout 52, Act: 2.160
2}Ġ=Ġ2014Ċ\]ĊRearrangingĠthis,ĠweĠget:Ċ\
Rollout 50, Act: 2.138
}{10}Ġ\right)^kĊ\]ĊĊSolvingĠthisĠgives:ĊĊ\[ĊkĠ=Ġ\
Rollout 97, Act: 2.112
ĠbeĠ\(y\).ĠThen,ĠtheĠtotalĠpressureĠisĠgivenĠby:ĊĠĠĠĠĠ\[ĊĠĠĠĠĠP_{\
Rollout 4, Act: 2.087
.ĠTheĠequivalentĠcapacitanceĠforĠcapacitorsĠinĠseriesĠisĠgivenĠby:ĊĊ\[ĊC_{\text{
Rollout 93, Act: 2.071
014}{2014}Ġ=Ġ1\),ĠweĠget:ĊĠĠĠĠĠ\[ĊĠĠĠĠ
Rollout 84, Act: 2.066
xĠ=Ġ4Ġ-Ġ4xĊ\]ĊĊ\[Ċ9Ġ-Ġ4Ġ=Ġ36
Rollout 2, Act: 2.043
timesĠ(-1)^{997}Ġ\equivĠ1Ġ\timesĠ3Ġ\timesĠ(-1
Rollout 76, Act: 2.035
^2Ġ=Ġ4\)Ġcolorings.ĠTotal:Ġ\(2Ġ\timesĠ4Ġ=Ġ8\
Rollout 76, Act: 2.031
8\).ĊĠĠĠ-ĠCombinedĠtotalĠforĠoverlappingĠpairs:Ġ\(32Ġ+Ġ8Ġ=Ġ40
Rollout 82, Act: 2.010
0Ġ+16Ġ+16)/26Ġ=Ġ42/26ĠâīĪ1.
Top 16 Negative Activations
Rollout 11, Act: -5.575
ĠsomeĠspecificĠcurves?ĠNotĠsure.ĊĊWait,ĠanotherĠthought.ĠSinceĠabcĠ=Ġ1,ĠtakeĠlogarithms
Rollout 30, Act: -5.578
,211Ġ/4,058,210.ĊĊTherefore,ĠtheĠreducedĠformĠwould
Rollout 88, Act: -5.579
0.ĠButĠlet'sĠcheck.ĊĊWait,ĠanotherĠpoint:ĠsometimesĠwhenĠpeopleĠaskĠforĠtheĠnumberĠofĠ"
Rollout 3, Act: -5.594
Ġm/s.ĠButĠneedĠtoĠspecify.ĊĊAlternatively,Ġhere'sĠanotherĠangle:ĠgasĠthermometricĠspeeds.ĠAt
Rollout 24, Act: -5.606
ĠcoveringĠanĠedgeĠinĠtheĠdominanceĠhypergraph.ĊĊAnotherĠthoughtĠinspiredĠbyĠlinearĠalgebra:ĠImagineĠcreatingĠaĠmatrixĠwhere
Rollout 80, Act: -5.625
ĠisĠthereĠaĠpartĠwhereĠIĠmessedĠupĠunits?ĠLet'sĠcheckĠunits:ĊĊEnergyĠperĠneutrino:Ġ
Rollout 32, Act: -5.630
Ġthere'sĠaĠpatternĠorĠaĠrecurrenceĠrelationĠhere.ĠLetĠmeĠthink.ĠMaybeĠfirstĠcomputeĠaĠfewĠpowersĠof
Rollout 3, Act: -5.649
161.4Ġm/s.ĊĊAsĠanĠassistant,ĠIĠneedĠtoĠperhapsĠsuggestĠbothĠpossibilities,Ġbut
Rollout 28, Act: -5.678
ĠderivedĠabove:Ġ4b2h2Ġ+Ġ4abĠhkĠ+4a2k2Ġ-
Rollout 22, Act: -5.762
ĠproceedĠinĠtheĠabsenceĠofĠthatĠinfo.ĊĊWaitĠanotherĠthought.ĠYouĠcanĠalsoĠseeĠthatĠinĠtextbooks,Ġwhile
Rollout 24, Act: -5.792
ĠsizeĠbyĠusingĠaĠpropertyĠthatĠisĠpreserved.ĊĊAnotherĠthought:ĠForĠeachĠminimalĠsaddleĠpairĠ(R,ĠC
Rollout 80, Act: -5.824
00ĠkmĠ=Ġ1.2ĠÃĹĠ10^8Ġcm.ĊĊVolumeĠ=Ġ(4/
Rollout 35, Act: -5.824
19ĠandĠ25:Ġnot.ĊĊTrialĠ3:ĠnâĪĴ17=-1,Ġn
Rollout 46, Act: -5.851
2Ġ/ĠÏģ2Ġ-ĠÏī2Ġ/Ġ2Ġ(x2Ġ+Ġy2)ĊĊThen:ĊĊ
Rollout 81, Act: -5.874
:ĊĊ20hĠ/Ġ(51Ġ-Ġh^2)Ġ=Ġ22/7Ġ->Ġcross
Rollout 98, Act: -5.946
ĠtheĠanswerĠchoicesĠonlyĠincludeĠoptionsĠAĠtoĠD.ĠLetĠmeĠconfirmĠeachĠsystemĠagain:ĊĊ1.ĠRejected

UP_PROJ

Top 16 Positive Activations
Rollout 30, Act: 3.368
ĠsumĠisĠsum_{n=1}^\inftyĠ[1/(a_{n+1}-1
Rollout 30, Act: 3.271
=1Ġ}^\inftyĠ1/(a_k(a_kĠ-1Ġ))ĠforĠtheĠsameĠreasonĠas
Rollout 30, Act: 3.263
ĠsameĠasĠSum_{k=1Ġ}^\inftyĠ1/(a_k(a_kĠ-1Ġ))
Rollout 1, Act: 3.254
x1Ġ-Ġx2,Ġ2y⣩Ġ=Ġ||x1||2Ġ-Ġ||x2||
Rollout 30, Act: 3.218
}^{\infty}Ġ\frac{b_nĠ(b_{n+2}Ġ-Ġb_n
Rollout 16, Act: 3.187
]Ġ=Ġsum_{m=0}^\inftyĠ[1Ġ-Ġ(1Ġ-Ġ(1/2
Rollout 11, Act: 3.181
)(ĠrĠe^{iθ})Ġ)Ġ=Ġr2Ġ+Ġ(1/r)Ġe^{-iθ}
Rollout 69, Act: 3.174
ĠsumĠSĠwouldĠtelescope:ĊĊS=ĠΣĠα_iĠ(Ġsec(theta_{i-1})Ġ-Ġsec
Rollout 66, Act: 3.152
nĠ-12nĠ*nĠ)=gcd(n,ĠnĠ)=n.ĊĊTherefore,ĠunlessĠn
Rollout 91, Act: 3.147
)ĠcanĠbeĠexpressedĠasĠ|uĠÃĹĠv|Ġ/Ġ|uĠâĭħĠvĠ|?ĠWait,
Rollout 96, Act: 3.147
098,815=Ġ-3,720,873ĊĊHence
Rollout 30, Act: 3.139
1Ġ+ĠSum_{k=1Ġ}^\inftyĠ[1/(a_kĠ-1Ġ)Ġ-Ġ
Rollout 36, Act: 3.121
Ġ-10,800Ġ=>Ġ-1,728ĊĊSo:ĊĊ2s2Ġ-
Rollout 1, Act: 3.115
ĠxĠandĠx',ĊĊ||xĠ-Ġx'||2Ġ=Ġd2,ĊĊandĠ⣨x,Ġx'⣩
Rollout 25, Act: 3.102
:ĊĊr_pĠ=Ġ(h2Ġ/Ġmu)Ġ/Ġ(1Ġ+ĠeĠ)ĊĊAsĠhĠ=Ġr*v
Rollout 18, Act: 3.099
=7aĠ+25Ġ=>Ġ7a=4-25=Ġ-21Ġ=>Ġa
Top 16 Negative Activations
Rollout 4, Act: -4.946
4)/2âīĪ6ĠpF.ĠWait,ĠbutĠwhenĠweĠdidĠtheĠpartialĠderivatives,ĠweĠgotĠ~
Rollout 54, Act: -4.953
Ġweaker,ĠthatĠmeansĠHSâĪĴĠisĠaĠstrongerĠbase.ĠWait,Ġwait.ĠNo:Ġwait,ĠtheĠlower
Rollout 44, Act: -4.955
ĠquestionĠis,ĠisĠthisĠtheĠmaximumĠpossibleĠwork?ĠButĠwait,ĠwhenĠcompressingĠirreversiblyĠagainstĠa
Rollout 53, Act: -4.966
ĠtoĠdouble-checkĠwhetherĠPĠisĠreallyĠtheĠcentroid.ĠButĠtheĠproblemĠstatesĠthatĠmediansĠADĠandĠCEĠintersect
Rollout 48, Act: -4.985
ĠtoĠgetĠ0x907F?ĠButĠthisĠisĠtheĠalternativeĠmethod.ĊĊWait,ĠperhapsĠthis
Rollout 95, Act: -5.017
,ĠsoĠmaybeĠtheĠratioĠkĠrelatesĠtoĠarea.ĠButĠaccordingĠtoĠtheĠpriorĠreasoning,ĠtheĠareaĠofĠtriangle
Rollout 39, Act: -5.050
321ĠonesĠfollowedĠbyĠaĠ0.ĠButĠinĠourĠcase,Ġwait,ĠbutĠ10
Rollout 44, Act: -5.066
ĠbeĠinĠanĠirreversibleĠprocessĠcomparedĠtoĠreversibleĠprocesses?ĠButĠaccordingĠtoĠtheĠpreviousĠexamples,Ġyes.ĠForĠexample
Rollout 27, Act: -5.098
ĪĠ0.866ĠkĠT.ĠButĠthat'sĠifĠEĠisĠ(1/2)
Rollout 86, Act: -5.111
/4Ġ+Ġ0=pi/4.ĊĊButĠwithĠtheĠpreviousĠsign:ĊĊThusĠreturning,ĠtheĠintegral
Rollout 44, Act: -5.151
ĠaĠhigherĠconstantĠpressure,ĠyouĠhaveĠmoreĠwork.ĠButĠifĠcompressingĠreversibly,ĠwhichĠisĠaĠslow
Rollout 74, Act: -5.173
ĠisĠtheĠcoefficientĠforĠtheĠheelingĠerrorĠcorrection.ĠButĠwait,ĠmaybeĠK_bĠandĠK_sĠhereĠare
Rollout 23, Act: -5.220
ĠreciprocalĠofĠtheĠdecayĠconstantĠisĠtheĠmeanĠlifetime.ĠButĠtheĠhalfĠlifeĠrelationsĠare.ĊĊButĠlet'sĠdouble
Rollout 41, Act: -5.242
,ĠifĠconsidered,ĠnumbersĠwithĠexactlyĠoneĠdigit.ĊĊButĠifĠweĠtakeĠallĠtheseĠsummedĠup:ĊĊ1-digit
Rollout 23, Act: -5.389
1/(sumĠofĠ1/T_i).ĠNo,Ġno.ĠTheĠdecayĠconstantsĠsum.ĊĊHangĠon,
Rollout 23, Act: -5.421
sumĠofĠ1/T_i).ĠNo,Ġno.ĠTheĠdecayĠconstantsĠsum.ĊĊHangĠon,Ġno.

DOWN_PROJ

Top 16 Positive Activations
Rollout 18, Act: 7.075
=7aĠ+25Ġ=>Ġ7a=4-25=Ġ-21Ġ=>Ġa
Rollout 64, Act: 6.018
ĠisĠinteger,ĠA=2*1Ġ-3=Ġ-1.ĠThenĠtheĠgeneralĠtermĠisĠa_n
Rollout 75, Act: 5.990
12Ġâī¡3Ġmod8ĠâĨĴĠaĠâī¡Ġ-9Ġâī¡Ġ-1Ġâī¡7Ġmod
Rollout 35, Act: 5.964
0.ĠSolutions:ĠbĠ=Ġ[-7Ġ±Ġsqrt(49Ġ-40)]/2Ġ=Ġ[-
Rollout 12, Act: 5.887
)Ġ=ĠK(k),ĠwithĠdk/dnuĠ=Ġ(Ġ-nuĠ/Ġsqrt(1Ġ-Ġnu^2
Rollout 75, Act: 5.869
ĠaĠ+Ġ20Ġâī¡3âĨĴĠaâī¡-17âī¡7Ġmod8.ĠNo.ĊĊ
Rollout 93, Act: 5.850
ĠformulaĠgivesĠsqrt(3/3.14âīĪ0.97ĠwhereasĠactualĠformulaĠgives33
Rollout 12, Act: 5.717
u)Ġ=ĠK(k),ĠwithĠdk/dnuĠ=Ġ(Ġ-nuĠ/Ġsqrt(1Ġ-Ġnu^
Rollout 19, Act: 5.513
Ġk=1...ĠthenĠn=1-5=Ġ-4ĠwhichĠisnâĢĻtĠpositive.ĠDisregard
Rollout 18, Act: 5.418
=7aĠ+15Ġ=>Ġ7a=8-15Ġ=Ġ-7Ġ=>Ġa=
Rollout 87, Act: 5.415
(1Ġ+Ġr2).ĠThenĠdu/drĠ=Ġ[Ġ(1/(2Ġsqrt(1Ġ+Ġr2
Rollout 87, Act: 5.353
âīĪ0.374,Ġ1/râīĪ2.67ĊĊSampleĠ2:Ġ(
Rollout 49, Act: 5.341
,ĠwhichĠisĠtrueĠinĠHausdorffĠspacesĠsinceĠpointsĠareĠclosed.ĠButĠthat'sĠnotĠhelpful.ĊĊ
Rollout 49, Act: 5.334
ĠknowĠthatĠinĠHausdorffĠspaces,ĠcompactĠsetsĠareĠclosed.ĠSoĠeveryĠcompactĠsubsetĠofĠXĠis
Rollout 64, Act: 5.317
2*1Ġ-3Ġ+1*2^n=Ġ-1Ġ+2^n.ĠForĠn=1
Rollout 75, Act: 5.310
Ġ16âī¡3Ġmod8âĨĴĠaĠâī¡-13âī¡3Ġmod8.ĠButĠa
Top 16 Negative Activations
Rollout 53, Act: -4.287
36=828?Ġ23*30=690,Ġ23
Rollout 46, Act: -4.299
Ġwritten?ĊĊWait,ĠperhapsĠsinceĠtheĠproblemĠmentionsĠexpressingĠthemĠ"inĠtermsĠofĠU",ĠtheyĠjustĠwantĠthe
Rollout 46, Act: -4.308
Ġthat.ĠHmm,ĠbutĠtheĠproblemĠsaysĠ"express...ĠinĠtermsĠofĠU".ĠIfĠUĠisĠgivenĠas
Rollout 93, Act: -4.353
=0Ġto1007,Ġ1008Ġterms.ĠApologies,ĠgettingĠtangledĠup
Rollout 46, Act: -4.432
tĊĊSimilarlyĠforĠAcc_y.ĊĊButĠtheĠproblemĠsaysĠ"inĠtermsĠofĠU".ĠIfĠweĠtakeĠU
Rollout 25, Act: -4.438
.118)âīĪĠ1.057ĊĊHenceĠdenominatorĠisĠ1+1
Rollout 46, Act: -4.442
ĠwouldĠbeĠtheĠgradients.ĊĊButĠagain,ĠtheĠquestionĠsaysĠ"ExpressĠd2x/dt2ĠandĠd
Rollout 46, Act: -4.451
Coriolis),ĠandĠtheĠproblemĠstatesĠtoĠexpressĠthemĠinĠtermsĠofĠU.ĠButĠunlessĠtheĠvelocityĠterms
Rollout 46, Act: -4.469
ĠofĠUĠandĠtheĠvelocities.ĠHowever,ĠtheĠproblemĠsaysĠ"inĠtermsĠofĠU",ĠwhichĠisĠaĠbit
Rollout 72, Act: -4.478
ĠevenĠn:ĠweĠcanĠtakeĠ1,1,-1ĠandĠsumĠto1.ĠButĠforĠoddĠn
Rollout 46, Act: -4.510
ĠleaveĠitĠatĠthat.ĠHmm,ĠbutĠtheĠproblemĠsaysĠ"express...ĠinĠtermsĠofĠU".ĠIfĠU
Rollout 30, Act: -4.529
2015*2014Ġ=2015*(2000Ġ+1
Rollout 46, Act: -4.632
Ġterms.ĠHowever,ĠsinceĠtheĠproblemĠonlyĠmentionsĠexpressingĠthemĠâĢľinĠtermsĠofĠUâĢĿ,ĠI'mĠnowĠbeginning
Rollout 82, Act: -4.921
.ĠTherefore,ĠtheĠproblemĠisĠequivalentĠtoĠsplittingĠtheĠsymbolsĠintoĠthreeĠgroups,ĠandĠforĠeachĠgroup,Ġcreating
Rollout 82, Act: -4.975
Ġbest.ĊĊSoĠtheĠapproachĠwouldĠbeĠtoĠpartitionĠtheĠsymbolsĠintoĠthreeĠgroupsĠsuchĠthatĠwhenĠeachĠgroupĠisĠHuffman
Rollout 82, Act: -5.958
'sĠthinkĠagain.ĊĊSupposeĠweĠpartitionĠtheĠsixĠsymbolsĠintoĠthreeĠgroups:ĠG1,ĠG2,

Layer 51

GATE_PROJ

Top 16 Positive Activations
Rollout 73, Act: 6.674
Ġ0Ġ=ĠsameĠexpression.ĊĊSimplify:ĊĊLet'sĠcomputeĠtheĠcoefficients:ĊĊ-Ġ(n+1)
Rollout 64, Act: 6.486
)/2Ġ*2^nĊĊSimplifying,Ġlet'sĠsubstituteĠn=2010:ĊĊa
Rollout 5, Act: 6.432
Ġ+Ġx))Ġ=Ġ125HĊĊLetĠmeĠfactorĠHĠout:ĊĊ(xĠ+Ġ17
Rollout 28, Act: 6.429
Ġ(2kĠ-b))Ġb2ĊĊLetĠmeĠfactorĠoutĠB:ĊĊBĠabĠ=ĠBĠ[Ġ-
Rollout 28, Act: 6.426
Ġ/Ġ(2kĠ-b))Ġb2ĊĊLetĠmeĠfactorĠoutĠB:ĊĊBĠabĠ=ĠBĠ[
Rollout 5, Act: 6.335
175)/(xĠ+Ġ75)ĊĊLetĠmeĠwriteĠtheĠentireĠequation:ĊĊ(xĠ+Ġ1
Rollout 95, Act: 6.312
Ġcomplicated.ĠMaybeĠcoordinateĠsystemĠisĠbetter.ĊĊLetĠmeĠplaceĠpointĠAĠatĠtheĠoriginĠ(0,0).
Rollout 11, Act: 6.267
i2θ}))(re^{iθ})ĊĊLetĠmeĠcomputeĠstepĠbyĠstep:ĊĊ-ĠaĠ+Ġb
Rollout 64, Act: 6.250
1Ġ)Ġ<=2^{2012}ĊĊLet'sĠsubstituteĠz=ĠxĠ-1,ĠsinceĠx
Rollout 84, Act: 6.235
/9)*(27/32).ĠLet'sĠcomputeĠ(2/9)*(27/3
Rollout 11, Act: 6.222
2θ}))(re^{iθ})ĊĊLetĠmeĠcomputeĠstepĠbyĠstep:ĊĊ-ĠaĠ+ĠbĠ=
Rollout 73, Act: 6.218
Ġ-Ġ0Ġ=ĠsameĠexpression.ĊĊSimplify:ĊĊLet'sĠcomputeĠtheĠcoefficients:ĊĊ-Ġ(n+1
Rollout 17, Act: 6.193
Ġ5.29Ġ*5.29:Ġ5^2=25,Ġ5
Rollout 95, Act: 6.179
ļ3ĠkĠ/490ĠâĪĪĠQ.ĠLetĠmeĠdenoteĠrĠ=ĠâĪļ3kĠ/4
Rollout 66, Act: 6.170
ĠtheĠlengthĠofĠthisĠaltitudeĠC1C2ĠwouldĠbeĠ(productĠofĠtheĠlegs)/hypotenuse.ĊĊ
Rollout 5, Act: 6.163
Ġ75)Ġ=Ġ125ĊĊLetĠmeĠletĠyĠ=ĠxĠ+Ġ75.
Top 16 Negative Activations
Rollout 37, Act: -4.000
ĠĠdividesĠĠ$\detĠA$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠjustĠcameĠacross
Rollout 62, Act: -4.010
ĠisĠ\(\boxed{132}\).<|im_end|>
Rollout 94, Act: -4.017
ĠcorrectĠanswerĠisĠ\(\boxed{B}\).<|im_end|>
Rollout 2, Act: -4.028
ĠisĠ\(\boxed{109}\).<|im_end|>
Rollout 44, Act: -4.042
FinalĠAnswer:Ġ\(\boxed{ABC}\)<|im_end|>
Rollout 29, Act: -4.043
FinalĠAnswer**Ċ\boxed{A}ĊĊ<|im_start|>answerĊGivenĠaĠgallonĠmilkĠjugĠwithĠaĠpluggedĠhandleĠdue
Rollout 57, Act: -4.045
.ĠčĊčĊ*A.ĠCsaszar*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshow
Rollout 91, Act: -4.047
|\)ĠisĠ\(\boxed{2}\).<|im_end|>
Rollout 11, Act: -4.065
\boxed{(-1,Ġ3)}ĊĊ<|im_start|>answerĊGivenĠthreeĠcomplexĠnumbersĠ\(a,Ġb,
Rollout 47, Act: -4.080
ĠWhatĠisĠtheĠvalueĠofĠ$N$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠthere'sĠthis
Rollout 4, Act: -4.084
ĠenergyĠstoredĠinĠthisĠcombinationĠofĠcapacitors?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠto
Rollout 28, Act: -4.132
a^2b^2Ġ=0}\)<|im_end|>
Rollout 95, Act: -4.136
**FinalĠAnswer**Ċ\boxed{0}ĊĊ<|im_start|>answerĊTriangleĠ\(ABC\)ĠhasĠsideĠlengthsĠ\(AB
Rollout 27, Act: -4.151
ĠcomponentĠofĠvelocityĠalongĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgot
Rollout 87, Act: -4.174
}$,ĠandĠanyĠnecessaryĠnumericalĠconstants.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfind
Rollout 48, Act: -4.217
ĠisĠvalid.Ġ\boxed{No}ĊĊ<|im_start|>answerĊToĠdetermineĠifĠtheĠIPv4ĠheaderĠcontainsĠanĠerror

UP_PROJ

Top 16 Positive Activations
Rollout 70, Act: 5.077
-digitĠnumbers,ĠweĠneedĠtoĠcheck.ĊĊButĠthisĠisĠgettingĠcomplicated.ĠMaybeĠinsteadĠofĠtryingĠm=4
Rollout 49, Act: 5.064
ĠaĠspaceĠwouldĠhaveĠotherĠproperties.ĠHmm.ĠThisĠseemsĠaĠbitĠvague.ĊĊWait,ĠletĠmeĠgetĠback
Rollout 5, Act: 5.013
ĠexceedingĠx2/100.ĠĊĊFirst,ĠletĠmeĠrecallĠsomeĠtrapezoidĠproperties.
Rollout 57, Act: 4.983
ĠofĠopenĠsetsĠnarrowingĠdownĠonĠC.ĠButĠthisĠisĠvague.ĠAlternatively,ĠconsiderĠthatĠinĠtheĠdensityĠtopology
Rollout 36, Act: 4.950
ĠfindĠtheĠdistanceĠfromĠPĠtoĠA.ĠĊĊFirst,ĠmaybeĠIĠshouldĠvisualizeĠtheĠcube.ĠLet'sĠassign
Rollout 30, Act: 4.931
ĠcomputeĠ2015Ġ*2014:ĊĊNoteĠthatĠ2014=2
Rollout 93, Act: 4.878
Ġ+1))ĠdependingĠonĠn.ĊĊWait,Ġthat'sĠtooĠintricate.ĊĊAlternatively,ĠanotherĠideaĠrelatedĠtoĠthe
Rollout 66, Act: 4.770
ĠFindĠ$p$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthroughĠthisĠproblem
Rollout 59, Act: 4.753
392ĠshareĠanyĠcommonĠfactors.ĊĊ123Ġfactors:Ġ3Ġ*Ġ41.ĊĊ3
Rollout 95, Act: 4.728
ĠthatĠhaveĠrationalĠlength.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠneedĠtoĠsolve
Rollout 10, Act: 4.716
Ġ2aĠ+Ġ3b,ĠbutĠthat'sĠnotĠhelpful.ĠWait,Ġ0ĠisĠeven,
Rollout 36, Act: 4.703
ĠFindĠ$AP.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠfigureĠoutĠhowĠto
Rollout 95, Act: 4.689
ĠsegmentsĠP_kQ_kĠhaveĠrationalĠlengths.ĊĊFirst,ĠletĠmeĠvisualizeĠtriangleĠABC.ĠLetĠmeĠsketchĠit
Rollout 3, Act: 4.684
ĠmolecularĠspeedĠofĠradonĠgas.ĠHmm,Ġfirst,ĠIĠrememberĠthere'sĠaĠformulaĠforĠtheĠrootĠmean
Rollout 93, Act: 4.674
/(k,1007).ĠWait,ĠthisĠlineĠofĠthinkingĠmightĠnotĠbeĠtooĠhelpful.ĊĊAlternatively
Rollout 15, Act: 4.654
ĠcanĠbeĠsuchĠthatĠallĠinitialĠstatesĠdieĠout.ĠButĠmaybeĠnot?ĊĊAlternatively,ĠperhapsĠtheĠgoodĠnsĠare
Top 16 Negative Activations
Rollout 17, Act: -4.376
.ĊĊCalculatingĠunits:ĊĊ(ĠT·m/AĠ*ĠJ/TĠ)Ġ/Ġm3Ġ=Ġ(J·
Rollout 14, Act: -4.389
ĠtheĠintegrals:ĊĊ-Ġ[I]Ġ=ĠâĪ«fĠdxĠ~Ġ[f]Ġ*Ġ[x],Ċ
Rollout 32, Act: -4.397
78ĊĊNowĠtheĠcoefficientsĠofĠÏī2:ĊĊ-670Ïī2Ġ+Ġ(-67
Rollout 3, Act: -4.429
s2·mol).ĠThen,ĠdividingĠbyĠMĠinĠkg/mol,ĠtheĠmolĠunitĠcancels,Ġthen
Rollout 89, Act: -4.464
ĠofĠtheĠchildrenĠareĠgirls.ĠLetĠthatĠparse.Ġ"2/5ĠofĠtheĠchildrenĠ(girls)"Ġrather
Rollout 14, Act: -4.515
]Ġ=ĠâĪ«fĠdxĠ~Ġ[f]Ġ*Ġ[x],Ċ-Ġ[J]Ġ=ĠâĪ«
Rollout 85, Act: -4.527
ĠcanĠtheyĠbeĠtheĠsame?ĠWait,ĠtheĠproblemĠstatesĠ"theĠsetĠofĠnumbersĠlabeledĠonĠtheĠpoints"
Rollout 41, Act: -4.529
Ġ9Ġ=Ġ81.ĠButĠtheĠproblemĠsaysĠatĠmostĠtwoĠdifferentĠdigits.ĠSinceĠaĠtwo-digit
Rollout 21, Act: -4.582
ĠinĠSĠhaveĠimageĠZ.ĠWait,Ġno:Ġ"imageĠexactlyĠZ"ĠmeansĠthatĠtheĠimageĠisĠthe
Rollout 23, Act: -4.587
ĠkeepingĠunitsĠinĠminutes.ĊĊComputeĠactivityĠinĠBqĠ(perĠsecond),ĠsoĠtimeĠt=10Ġminutes
Rollout 26, Act: -4.598
ĠtwoĠdecimalĠplaces.ĠWaitĠtheĠunitsĠinĠtheĠproblemĠsaysĠ'XĠkJĠmol^âĪĴ1',Ġand
Rollout 46, Act: -4.643
ĠofĠUĠandĠtheĠvelocities.ĠHowever,ĠtheĠproblemĠsaysĠ"inĠtermsĠofĠU",ĠwhichĠisĠaĠbit
Rollout 69, Act: -4.736
ĠtheĠsumĠcanĠbeĠwrittenĠasĠ(1/4)(sec(theta_{i-1})Ġ-Ġsec(theta
Rollout 88, Act: -4.805
104.ĠButĠtheĠwordingĠhereĠisĠonlyĠ"weights",ĠwhichĠtypicallyĠrefersĠtoĠtheĠmultiplicativeĠparameters
Rollout 69, Act: -4.820
)Sum_{i=0}^\inftyĠ(sec(theta_{i-1})Ġ-Ġsec(theta
Rollout 69, Act: -4.903
ĠSĠwouldĠtelescope:ĊĊS=ĠΣĠα_iĠ(Ġsec(theta_{i-1})Ġ-Ġsec(theta

DOWN_PROJ

Top 16 Positive Activations
Rollout 62, Act: 5.728
ĠstartingĠfromĠ\(ĠmĠ=Ġ1Ġ\)ĠandĠmovingĠupwardsĠuntilĠweĠfindĠtheĠdesiredĠnumberĠmakesĠsense.ĊĊ
Rollout 42, Act: 5.625
ĠareĠtreatedĠwithĠacid.ĠPhosphoricĠacidĠisĠaĠstrongĠacid,ĠandĠinĠsuchĠconditions,Ġalcoh
Rollout 42, Act: 5.544
olsĠareĠtreatedĠwithĠacid.ĠPhosphoricĠacidĠisĠaĠstrongĠacid,ĠandĠinĠsuchĠconditions,Ġal
Rollout 81, Act: 5.352
0.ĊĊComputeĠterms:ĊĊ484x2Ġ-x(533*102Ġ+
Rollout 42, Act: 5.342
atesĠtheĠhydroxylĠgroup,ĠconvertingĠitĠintoĠaĠgoodĠleavingĠgroupĠ(water).ĠThen,ĠtheĠelimination
Rollout 42, Act: 5.229
ĠTheĠhydroxylĠgroupĠisĠprotonated,ĠformingĠaĠgoodĠleavingĠgroupĠ(water).ĠUponĠdepartureĠofĠwater
Rollout 30, Act: 5.208
15*(2000Ġ+14)=2015*2000Ġ+
Rollout 75, Act: 5.197
.n^2Ġ-Ġ(n-1)^2=2n-1.ĠButĠforĠoddĠsquares,
Rollout 42, Act: 5.189
ocation.ĠThen,ĠtheĠbaseĠ(waterĠactingĠasĠaĠweakĠbase)ĠwouldĠdeprotonateĠaĠnearby
Rollout 42, Act: 5.140
propylĠring.ĠWhenĠtheĠacidĠisĠadded,ĠitĠprotonatesĠtheĠhydroxylĠgroup,ĠconvertingĠit
Rollout 63, Act: 5.134
f(1,Ġ1)Ġ=Ġ1Ġ+Ġ0Ġ+Ġ0Ġ=Ġ1Ċ\
Rollout 44, Act: 5.108
abaticĠexpansion,ĠpressureĠdropsĠmore).ĠTherefore,ĠareaĠunderĠisothermalĠcurveĠisĠlarger;ĠsoĠworkĠdone
Rollout 42, Act: 5.006
ĠinĠaqueousĠH3PO4,ĠwhichĠisĠaĠgoodĠacid,Ġit'sĠmoreĠlikelyĠanĠE1
Rollout 75, Act: 4.931
Ġm2Ġ=Ġ(m-1)2+Ġ(difference)ĠwhichĠisĠeven?ĠSoĠ(m-
Rollout 42, Act: 4.926
lopropylĠring.ĠWhenĠtheĠacidĠisĠadded,ĠitĠprotonatesĠtheĠhydroxylĠgroup,Ġconverting
Rollout 75, Act: 4.922
ĠasĠm2Ġ=Ġ(m-1)2+Ġ(difference)ĠwhichĠisĠeven?ĠSoĠ(m
Top 16 Negative Activations
Rollout 2, Act: -4.251
ĠimpossibleĠbecauseĠtheĠnumbersĠareĠhuge.ĠSo,ĠperhapsĠthere'sĠaĠpatternĠorĠperiodicĠbehaviorĠweĠcanĠexploitĠwhen
Rollout 95, Act: -4.253
k/2450=5*sqrt(3)*k/2450.ĊĊThus
Rollout 51, Act: -4.276
ĠbeĠremovedĠbyĠadjustingĠtheĠsignĠappropriately.ĊĊLetĠmeĠrecallĠthatĠtheĠabsoluteĠvalueĠofĠsin(xĠ-Ġθ
Rollout 16, Act: -4.308
ButĠcalculatingĠthisĠsumĠdirectlyĠmightĠbeĠchallenging.ĠSoĠperhapsĠweĠcanĠuseĠinclusion-exclusionĠorĠsomeĠcombinator
Rollout 72, Act: -4.334
ĠtoĠ1.ĠĊĊFirst,ĠIĠneedĠtoĠrecallĠthatĠtheĠrootsĠofĠunityĠofĠorderĠ\(ĠnĠ\
Rollout 23, Act: -4.369
57)^7.ĠHmm,Ġimpractical.ĠPerhapsĠuseĠapproximation.ĊĊAlternatively,ĠmaybeĠuseĠtheĠnaturalĠlogarith
Rollout 87, Act: -4.398
ĠperhapsĠthisĠwouldĠgetĠtooĠinvolved.ĊĊAlternatively,ĠperhapsĠtryĠnumericalĠintegrationĠforĠtheĠdimensionlessĠconstant.ĠSinceĠthe
Rollout 91, Act: -4.421
ĠofĠtriangle).ĠWait,ĠnotĠexactly.ĠLet'sĠrecallĠthatĠinĠtriangle,ĠcotĠBĠ=Ġadjacent/opposite
Rollout 21, Act: -4.437
Ġx_i;ĠyouĠcannotĠquantifyĠexistentially.ĠUnlessĠthere'sĠaĠwayĠtoĠexpressĠsurjectivity.ĊĊWait
Rollout 11, Act: -4.516
)Ġ)Ġ]ĊĊButĠthisĠisĠveryĠcomplicated,ĠperhapsĠthere'sĠnoĠimmediateĠsimplification.ĠHowever,ĠifĠI
Rollout 97, Act: -4.534
ĠtotalĠvolumeĠ(concentrationĠtimesĠRT?),ĠbutĠperhapsĠsimplerĠtoĠrelateĠviaĠtheĠstoichiometry.ĊĊAlternatively,
Rollout 40, Act: -4.577
ĠOrĠperhapsĠusingĠcomplexĠnumbers?ĠHmm.ĠLetĠmeĠrecallĠthatĠrotationsĠcanĠbeĠrepresentedĠusingĠcomplexĠnumbersĠbyĠmultiplying
Rollout 16, Act: -4.621
ling,ĠmultipleĠconversionsĠpossible.ĊĊWait,Ġlet'sĠcheckĠaĠsample.ĠWhat'sĠtheĠexpectationĠuntilĠoneĠsap
Rollout 93, Act: -4.639
ĠMyĠgoalĠisĠtoĠevaluateĠthisĠsum.ĠLetĠmeĠthinkĠaboutĠhowĠtoĠapproachĠit.ĠĊĊFirstĠoff,
Rollout 66, Act: -4.983
ĠtoĠfindĠtheĠperimeterĠp.ĊĊAlright,ĠletĠmeĠrecallĠthatĠinĠaĠright-angledĠtriangle,ĠtheĠaltitude
Rollout 11, Act: -5.175
ĠyĠ)ĊĊButĠnowĠthisĠseemsĠreallyĠcomplicated.ĠPerhapsĠthere'sĠanotherĠsubstitution?ĊĊAlternatively,ĠsettingĠvariablesĠinĠterms

Layer 52

GATE_PROJ

Top 16 Positive Activations
Rollout 39, Act: 2.076
Ġ+Ġ9999Ġ+Ġ\cdotsĠ+Ġ\underbrace{99\ldotsĠ
Rollout 44, Act: 2.040
=\mathrm{T}_{2}$,ĠandĠ(ii)Ġpositive,ĠifĠitĠisĠexpandedĠreversiblyĠunderĠad
Rollout 91, Act: 1.919
(\overrightarrow{CB}Ġ=Ġ(-2,Ġ0)\).ĊĊ3.Ġ**Calculations**:Ċ
Rollout 17, Act: 1.896
\mathbf{I}=\intĠ\mathbf{B}(\mathbf{r})|\
Rollout 12, Act: 1.713
x^2)}}\]čĊ(whereĠtheĠpositiveĠvalueĠofĠtheĠsquareĠrootĠisĠtaken)ĠisĠmonotonically
Rollout 12, Act: 1.703
^2)}}\]čĊ(whereĠtheĠpositiveĠvalueĠofĠtheĠsquareĠrootĠisĠtaken)ĠisĠmonotonicallyĠdecreasing
Rollout 75, Act: 1.674
Ġ+Ġ9\)ĠandĠ\(9Ġ=Ġ4Ġ+Ġ4Ġ+Ġ1\).ĊĊ4.Ġ**
Rollout 45, Act: 1.598
{}Ġ\minus{}4$ĠĠandĠĠ$Ġf(x\plus{}y)Ġ\equal{}Ġf
Rollout 33, Act: 1.582
$Ġ$\leq$Ġ$k$Ġ$\leq$Ġ$5$Ġ,ĠatĠleastĠoneĠofĠthe
Rollout 12, Act: 1.576
^2x^2)}}\]čĊ(whereĠtheĠpositiveĠvalueĠofĠtheĠsquareĠrootĠisĠtaken)ĠisĠmon
Rollout 44, Act: 1.508
,ĠandĠ(ii)Ġpositive,ĠifĠitĠisĠexpandedĠreversiblyĠunderĠadiabaticĠconditionsĠwithĠ$\
Rollout 30, Act: 1.494
\right)$ĠandĠ$b_{n+1}=a_{n}Ġb_{n-1}-
Rollout 2, Act: 1.468
Ġ999Ġ\timesĠ\cdotsĠ\timesĠ\underbrace{99\cdots9
Rollout 28, Act: 1.440
ĠtheĠconicĠequation,Ġ\(B^2Ġ-Ġ4AC\),ĠisĠcalculatedĠtoĠbeĠpositiveĠfor
Rollout 2, Act: 1.435
timesĠ\cdotsĠ\timesĠ\underbrace{99\cdots9}_{\text{9
Rollout 44, Act: 1.407
ĠworkĠdoneĠonĠtheĠgasĠisĠmaximumĠwhenĠitĠisĠcompressedĠirreversiblyĠfromĠ(p2,ĠV2
Top 16 Negative Activations
Rollout 32, Act: -7.298
Ġthere'sĠaĠpatternĠorĠaĠrecurrenceĠrelationĠhere.ĠLetĠmeĠthink.ĠMaybeĠfirstĠcomputeĠaĠfewĠpowersĠof
Rollout 46, Act: -7.315
ĠorĠsomeĠcombination,ĠbutĠnotĠsure.ĊĊWait,Ġlet'sĠgoĠback.ĠTheĠproblemĠsays:ĊĊ"Express
Rollout 55, Act: -7.325
n,Ġ0ĠwhenĠraisedĠtoĠtheĠpower.ĠLetĠmeĠcheckĠthat.ĠIfĠIĠthinkĠaboutĠAĠsquared
Rollout 58, Act: -7.344
ĠseemsĠcomplicated,ĠbutĠperhapsĠweĠcanĠsimplifyĠit.ĠLet'sĠnoteĠthatĠP(0)ĠisĠtheĠproduct
Rollout 23, Act: -7.366
,ĠmaybeĠtheĠuserĠdidn'tĠspecifyĠunitsĠconversion.ĠLetĠmeĠcheckĠagainĠthoroughly.ĊĊLet'sĠrecastĠall
Rollout 67, Act: -7.393
ĠfindingĠtheĠgeneralĠquadraticĠterm,ĠthenĠfindĠterms.ĠLet'sĠproceedĠbothĠways.ĊĊFirstĠmethod:ĠassumeĠthe
Rollout 98, Act: -7.404
ĠtheĠanswerĠchoicesĠonlyĠincludeĠoptionsĠAĠtoĠD.ĠLetĠmeĠconfirmĠeachĠsystemĠagain:ĊĊ1.ĠRejected
Rollout 95, Act: -7.430
,ĠseemsĠcomplicated.ĠMaybeĠcoordinateĠsystemĠisĠbetter.ĊĊLetĠmeĠplaceĠpointĠAĠatĠtheĠoriginĠ(0,
Rollout 87, Act: -7.439
Ġ]ĠdsĠdtĠ*ĠaĊĊWait,Ġno.ĠLet'sĠcheckĠagain:ĊĊOriginalĠexpression:ĊĊStartingĠfromĠV
Rollout 46, Act: -7.447
U/âĪĤx,ĠmaybeĠwithĠaĠsign.ĠLetĠmeĠcheck.ĠIfĠUĠisĠtheĠpotentialĠenergy,
Rollout 66, Act: -7.449
.ĠEachĠaltitudeĠlengthĠisĠmultipliedĠbyĠaĠfactor.ĠLet'sĠcheck.ĊĊFirstĠaltitudeĠ(fromĠC0):
Rollout 9, Act: -7.454
ĠI'mĠconfused.ĠWaitĠsquaresĠhaveĠfourĠsides.ĠLetĠmeĠactuallyĠassignĠcoordinatesĠtoĠtheĠsquareĠtoĠmakeĠthis
Rollout 73, Act: -7.474
0.ĠThen,ĠperhapsĠcomputeĠtheĠfirstĠsum.ĠLetĠmeĠcomputeĠthat.ĊĊLetĠmeĠdenoteĠS1Ġ=
Rollout 30, Act: -7.490
ThatĠtelescopesĠifĠweĠwriteĠitĠasĠsuch.ĠLet'sĠcheck:ĊĊSumĠfromĠn=1ĠtoĠâĪ
Rollout 43, Act: -7.503
ĠwhereĠmaybeĠthere'sĠaĠdifferentĠarrangement.ĊĊWait,ĠletĠmeĠcheckĠagainĠtheĠanswerĠoptionsĠasĠgiven:ĊĊAnswer
Rollout 30, Act: -7.979
ĠledĠusĠtoĠaĠmoreĠhandle-ableĠform.ĊĊLetĠmeĠrecap:ĊĊSo,ĠtheĠoriginalĠtermĠinĠthe

UP_PROJ

Top 16 Positive Activations
Rollout 52, Act: 6.919
:Ġ(-501+505)/2=4/2=2.ĠSoĠsumĠis
Rollout 52, Act: 6.530
2Ġ=Ġ(1007Ġ-3)/2=1004/2=50
Rollout 52, Act: 6.464
Ġ(4028Ġ-1Ġ+1)/2Ġ=4028/2=20
Rollout 68, Act: 6.428
Ġx2=Ġ(9Ġ-4Ġ-3)/2=2/2=1.ĠSoĠbothĠx
Rollout 52, Act: 6.363
)/2=Ġ(76Ġ-52)/2=24/2=12.ĠSo
Rollout 99, Act: 6.324
.14794)/0.25=0.05103/0
Rollout 52, Act: 6.311
2=Ġ(212Ġ-18)/2=194/2=97.
Rollout 81, Act: 6.273
=(Ġ-70Ġ+172)/22=102/22=51
Rollout 45, Act: 6.265
Ġ-4=Ġ-64Ġ+64Ġ-4=Ġ-4ĠâľĶï¸ı.ĠOkay,Ġmatches.ĊĊ
Rollout 68, Act: 6.226
Ġ[-(4Ġ-Ġ8)]Ġ/Ġ2Ġ=Ġ[4]Ġ/2Ġ=Ġ2.ĠSo
Rollout 52, Act: 6.219
028/1Ġ-1Ġ+1)/2Ġ=Ġ(4028Ġ-1Ġ+1
Rollout 99, Act: 6.126
2.19897)/0.5âīĪ-0.19897/
Rollout 82, Act: 6.013
10Ġ+16Ġ+16)/26Ġ=Ġ42/26ĠâīĪ1
Rollout 75, Act: 5.967
ĠsoĠ(1Ġ+1Ġ+1-3)/4=Ġ(3-3)/4=0mod
Rollout 99, Act: 5.914
19897)/0.5ĠâīĪĠ(-0.19897)/0
Rollout 0, Act: 5.914
Ġ2,3,5,7ĠareĠeven?ĊĊWaitĠfloorĠoperations:ĊĊForĠprimeĠ2:Ġ
Top 16 Negative Activations
Rollout 30, Act: -4.692
ĠthatĠtheĠsequencesĠareĠdefinedĠrecursivelyĠwithĠoverlappingĠterms,ĠmaybeĠthere'sĠaĠpatternĠhereĠorĠaĠwayĠtoĠexpress
Rollout 93, Act: -4.703
,ĠandĠwe'reĠsummingĠoverĠmanyĠterms,ĠmaybeĠweĠcanĠapproximateĠeachĠtermĠusingĠLaplace'sĠmethod
Rollout 82, Act: -4.708
?ĠThat'sĠtheĠcrux.ĠLetĠmeĠthinkĠifĠthere'sĠaĠsystematicĠway.ĊĊOneĠheuristicĠisĠto
Rollout 14, Act: -4.721
ĠtermsĠofĠ$J$ĠandĠ$K$.ĊĊMaybeĠCauchy-SchwarzĠonĠ$I$
Rollout 40, Act: -4.760
Ġorigin,ĠandĠthenĠtheĠtranslationĠisĠaĠshift,ĠmaybeĠweĠcanĠrepresentĠthisĠasĠaĠcombinationĠofĠrotationĠand
Rollout 69, Act: -4.835
ĠiĠincreases,ĠbothĠanglesĠgoĠtoĠzero.ĠSoĠperhapsĠweĠcanĠapproximateĠtermsĠforĠlargeĠi?ĠHowever,
Rollout 69, Act: -4.851
ĠtheĠanglesĠinvolveĠdivisionsĠbyĠ3^i,ĠperhapsĠtheĠsumĠhasĠsomeĠrecursiveĠrelation.ĠForĠexample,
Rollout 34, Act: -4.932
ĠconvolutionĠkernelĠbecauseĠitĠdependsĠonĠtĠ-Ġs.ĠMaybeĠIĠcanĠconvertĠtheseĠintegralĠequationsĠintoĠdifferentialĠequationsĠusing
Rollout 33, Act: -4.964
angementsĠbutĠgeneralizedĠtoĠdifferentĠpositions.ĊĊAlternatively,ĠmaybeĠIĠcanĠmodelĠthisĠproblemĠusingĠinclusion-exclusion,
Rollout 75, Act: -5.007
Ġeasier.ĠHowever,ĠsinceĠsixĠsquaresĠisĠallowed,ĠperhapsĠusingĠLagrange'sĠtheoremĠasĠaĠbasis,Ġthen
Rollout 69, Act: -5.013
(3Ġtheta_i)Ġ]ĊĊWeĠhaveĠtoĠseeĠifĠeachĠtermĠcanĠbeĠrewrittenĠinĠtermsĠofĠaĠtelesc
Rollout 69, Act: -5.022
)/cos(3Ġtheta_i).ĠLet'sĠseeĠifĠthisĠcanĠbeĠmanipulated.ĊĊPerhapsĠuseĠtheĠidentityĠfor
Rollout 69, Act: -5.047
ĠneedĠanĠexactĠvalue.ĊĊAlternatively,Ġlet'sĠseeĠifĠweĠcanĠrewriteĠtheĠtermĠinĠtermsĠofĠtangentĠor
Rollout 77, Act: -5.094
ĠthirdĠproperty,ĠwhichĠseemsĠlikeĠaĠfunctionalĠequation.ĠMaybeĠIĠcanĠuseĠthisĠtoĠreduceĠf(14
Rollout 63, Act: -5.164
Ġ\(Ġf(p,Ġq)Ġ\).ĠAlternatively,ĠperhapsĠexpressĠitĠinĠtermsĠofĠ\(ĠpĠ\)Ġand
Rollout 83, Act: -5.273
ĠseemsĠlikeĠaĠlinearĠrecurrenceĠrelation.ĠLetĠmeĠseeĠifĠthere'sĠaĠpatternĠhere.ĠIfĠweĠhaveĠthe

DOWN_PROJ

Top 16 Positive Activations
Rollout 90, Act: 6.503
.ĊĊFirst,ĠcomputeĠhowĠmuchĠ2018ĠdividesĠinto10,080,0
Rollout 20, Act: 6.331
46ĠdividesĠ966?Ġ46ĠdividesĠintoĠ966?Ġ46
Rollout 62, Act: 6.116
:Ġ7*2075=14525,Ġ14531
Rollout 16, Act: 6.105
ĠdividedĠbyĠ217.Ġ217*5=1085,ĠsoĠ
Rollout 70, Act: 6.005
:Ġ21.857.Ġ7*21=147,Ġremainder6
Rollout 96, Act: 5.944
7Ġby2197:ĊĊ2197*31=2197*3
Rollout 87, Act: 5.937
(r,θ)Ġ[Ġ(1/r)Ġ*Ġr2ĠsinθĠdrĠdθĠdÏĨĠ]ĊĊBut
Rollout 95, Act: 5.759
???ĠWait.ĊĊWait,ĠholdĠon.ĠLetĠmeĠrecastĠthis.ĊĊPointĠPkĠisĠalongĠAB.
Rollout 71, Act: 5.704
57...ĠwhichĠisĠnotĠanĠinteger.Ġ11?Ġ2011Ġ÷1
Rollout 90, Act: 5.684
Ġlet'sĠcomputeĠhowĠmanyĠtimesĠ2018ĠgoesĠintoĠ336,000
Rollout 62, Act: 5.589
'sĠtestĠdivideĠbyĠ17:Ġ17*59=1003?Ġ
Rollout 20, Act: 5.525
6ĠdividesĠintoĠ966?Ġ46Ġ*21Ġ=Ġ966Ġ(4
Rollout 65, Act: 5.517
2ĠifĠandĠonlyĠifĠa2ĠhasĠinĠitsĠprimeĠfactorsĠ3,5,7,19
Rollout 98, Act: 5.479
ĠhaveĠaĠshorterĠmain-sequenceĠtime.ĠLetĠmeĠrecalculate:Ġmain-sequenceĠlifetimeĠformula:ĠtĠ~
Rollout 99, Act: 5.469
ĠwouldĠgiveĠx4.ĠButĠifĠtheĠuserĠisĠboundĠtoĠdoĠtwoĠiterations,ĠwouldĠtheĠanswerĠbeĠx
Rollout 37, Act: 5.469
ĠsoĠwhenĠreducingĠmoduloĠp,ĠtheĠmatrixĠisĠoverĠtheĠfieldĠF_p.ĠNow,ĠtheĠproblemĠreducesĠto
Top 16 Negative Activations
Rollout 78, Act: -5.699
ĠthatĠisĠ110/20Ġ=Ġ11/2Ġ=Ġ5.5Ġminutes
Rollout 3, Act: -5.717
0ĊĊsqrt(27,950)=ĠapproximatelyĠ167.2Ġm/s.
Rollout 56, Act: -5.821
1âĪļ3Ġ/12)^2Ġ=Ġ(121*3)/144Ġ=Ġ
Rollout 3, Act: -5.834
225,753Ġ)ĠâīĪĠ475Ġm/s.ĠWhichĠisĠcorrect.ĊĊ
Rollout 81, Act: -5.904
9584)Ġ=Ġlet'sĠcheck:Ġ172^2=2958
Rollout 3, Act: -5.963
ĠLet'sĠsee.ĊĊ168^2Ġ=Ġ(170Ġ-Ġ2)^2Ġ=Ġ
Rollout 23, Act: -5.970
-lifeĠ(sinceĠtheĠbetaĠdecayĠhalf-lifeĠisĠonlyĠ0.098Ġmin,ĠwhichĠisĠthe
Rollout 78, Act: -6.018
60Ġ=Ġ22/40Ġ=Ġ11/20Ġminutes.ĊĊSimilarly,Ġthe
Rollout 3, Act: -6.026
(28408.12)=Ġ168.546...ĠHence,
Rollout 3, Act: -6.100
28,408.12)=Ġ168.5ĊĊThus,ĠanswerĠis
Rollout 56, Act: -6.211
Ġ]Ġ=Ġsqrt[49/36Ġ+Ġ(121*3)/144Ġ]
Rollout 69, Act: -6.254
1938,ĠsquaredĠâīĪĠ0.000376.Ġcos(3
Rollout 56, Act: -6.289
11âĪļ3Ġ/12)^2Ġ=Ġ(121*3)/144Ġ=
Rollout 3, Act: -6.536
ĠThenĠsqrt(225,298)=ĠapproxĠ474.6Ġm/s.
Rollout 69, Act: -6.800
.01938,ĠsquaredĠâīĪĠ0.000376.Ġcos
Rollout 3, Act: -6.943
(225,298)=ĠapproxĠ474.6Ġm/s.ĠSoĠabout

Layer 53

GATE_PROJ

Top 16 Positive Activations
Rollout 8, Act: 7.357
*9.Ġ31ĠandĠ126ĠareĠcoprime,ĠsoĠyeah,Ġsimplified.ĊĊ
Rollout 61, Act: 7.338
3ĠandĠ780ĠareĠcoprime.ĠLet'sĠcheck.ĊĊDivisorsĠofĠ13
Rollout 47, Act: 7.328
ĠfoundĠv_airĠ=Ġ(4/3)Ġvt.ĠLet'sĠsubstitute:ĊĊp0Ġ-ĠpĠ=Ġ(
Rollout 58, Act: 7.247
018)(2019)/2.ĠCalculatingĠthat:Ġ2018Ġ*
Rollout 72, Act: 7.233
Ġ1)/4Ġ*Ġ2?ĠLetĠmeĠverify:ĊĊWeĠknowĠthatĠcos(36°)
Rollout 16, Act: 7.223
atorsĠareĠ217ĠandĠ63.ĠFactoring:ĊĊ217Ġ=Ġ7*
Rollout 16, Act: 7.195
Ġ7880ĠandĠ1953ĠareĠcoprimeĠasĠestablishedĠearlier,ĠsoĠthe
Rollout 20, Act: 7.178
7*23*43*47.ĠToĠcheck:ĊĊTheĠLCMĠtakesĠtheĠmaxĠex
Rollout 20, Act: 7.176
ĠcommonĠmultipleĠofĠ42ĠandĠ46.ĠLetĠmeĠcomputeĠthat.ĊĊ42ĠfactorsĠintoĠ
Rollout 71, Act: 7.165
92+492Ġ+102.ĠLetĠmeĠcomputeĠthat:ĊĊ392=Ġ
Rollout 9, Act: 7.148
ĊĊSoĠ20yĠ=Ġ100Ġ=>ĠyĠ=Ġ5.ĊĊTherefore,ĠtheĠy
Rollout 11, Act: 7.131
/(ab))Ġ+Ġ(1/(ab))/a.ĠSimplifyĠeachĠterm:Ġa/bĠisĠtheĠsame
Rollout 38, Act: 7.092
.ĠWait,Ġ196/672=Ġ(28*7)/(28*
Rollout 77, Act: 7.082
Ġ(52/38)*266.ĠCalculateĠthat:Ġ52/38Ġsimpl
Rollout 17, Act: 7.078
.29ĠÃĹ10^-11Ġm.ĠTherefore,Ġa03=(5.29
Rollout 16, Act: 7.077
Ġ7880/1953.ĊĊLetĠmeĠcheckĠifĠ7880Ġand
Top 16 Negative Activations
Rollout 25, Act: -1.494
649e7)^2Ġ=Ġ7.479e15ĊĊThenĠnumerator:
Rollout 33, Act: -1.532
4,Ġ5,Ġ6$ĠsuchĠthatĠforĠeachĠ$k$ĠwithĠ$1$Ġ$\le
Rollout 48, Act: -1.547
ĠtreatingĠthemĠasĠunsignedĠ16-bit:Ġ46,610.ĠAddingĠ17
Rollout 48, Act: -1.549
Ġvalue)ĠinĠtheĠIPĠheader:Ġ4500Ġ0034ĠB612
Rollout 48, Act: -1.567
64).ĊĊ0x5BC6ĠisĠ23444Ġ(approx:Ġ5*
Rollout 23, Act: -1.602
0.098}Ġ\approxĠ7.0724Ġ\,Ġ\text{min
Rollout 80, Act: -1.667
Ġ10^24ĠâīĪĠ7.238ĠÃĹĠ10^24
Rollout 17, Act: -1.711
5.29*5.29=27.9841,ĠthenĠ2
Rollout 38, Act: -1.741
5/124ĠxâīĪ1.25x.ĊĊSoĠLengthĠLâīĪ1.25
Rollout 38, Act: -1.764
155/124âīĪ1.25.ĠSo,Ġx<=1.25
Rollout 72, Act: -1.788
cos(72°)ĠâīĪĠ0.618,ĠsoĠ1Ġ+Ġ0
Rollout 44, Act: -1.854
ĊĊ[D]ĠTheĠchangeĠinĠinternalĠenergyĠofĠtheĠgasĠisĠ(i)Ġzero,ĠifĠitĠisĠexpanded
Rollout 86, Act: -1.880
5708)^5Ġ/5ĠâīĪĠ8.89Ġ/5ĠâīĪĠ1
Rollout 38, Act: -2.103
55/124ĠxâīĪ1.25x.ĊĊSoĠLengthĠLâīĪ1.2
Rollout 78, Act: -2.124
ĠalongĠaĠpathĠthatĠparallelsĠtwoĠside-by-sideĠtrainĠtracksĠrunningĠtheĠeast/westĠdirection.ĠJonĠridesĠeast
Rollout 95, Act: -2.213
},ĠP_{1},ĠP_{2},Ġ...Ġ,ĠP_{2450}Ġ=ĠB$

UP_PROJ

Top 16 Positive Activations
Rollout 25, Act: 4.466
ĠhyperbolicĠtrajectory:ĊĊeĠ=Ġsqrt(1Ġ+Ġ(2ĠεĠh2)/mu2)Ġ)ĊĊ
Rollout 22, Act: 4.400
6.02214076Ġ=ĠapproximatelyĠ0.3347,Ġthen
Rollout 25, Act: 4.195
Ġ/Ġμ)Ġ/Ġ(1Ġ+Ġsqrt(1Ġ+Ġ(2Ġ*ĠεĠ*Ġh2)/μ2
Rollout 27, Act: 4.146
Ġgas,ĠT_cĠisĠgivenĠby:ĊĊT_cĠ=Ġ(Ġ(n^(2/3)Ġh^
Rollout 25, Act: 4.087
/2ĊĊThen,ĠeĠ=Ġsqrt(1Ġ+Ġ2ĠepsilonĠh2Ġ/mu2Ġ)ĊĊYes
Rollout 25, Act: 3.996
Ġ/Ġμ)Ġ/Ġ(1Ġ+Ġsqrt(1Ġ+Ġ2*ε*h2/μ2))ĊĊ
Rollout 25, Act: 3.869
Ġ*Ġ1Ġ/Ġ[1Ġ+Ġsqrt(1Ġ+Ġ(2ĠepsilonĠh2)/mu2Ġ+Ġ
Rollout 27, Act: 3.868
ĠanotherĠapproachĠmightĠbeĠtoĠuseĠthatĠkĠT_cĠ=Ġ(1/2)ĠmĠv_rms^
Rollout 25, Act: 3.778
Ġtrajectory,ĠtheĠperigeeĠis:ĊĊr_pĠ=Ġ(h2Ġ/Ġμ)Ġ/Ġ(1Ġ+
Rollout 25, Act: 3.681
2/2ĊĊThen,ĠeĠ=Ġsqrt(1Ġ+Ġ2ĠepsilonĠh2Ġ/mu2Ġ)ĊĊ
Rollout 25, Act: 3.658
.ĊĊTheĠspecificĠorbitalĠenergyĠεĠisĠgivenĠbyĠεĠ=Ġv2Ġ/Ġ2Ġ-ĠμĠ/Ġr.ĊĊ
Rollout 27, Act: 3.634
_rmsĠ=Ġ5.54ĠÃĹĠ10^-17ĠJ.ĊĊWeĠneedĠtoĠfind
Rollout 25, Act: 3.633
ĠisĠpositive,ĠεĠ=Ġv2Ġ/Ġ2Ġ-ĠμĠ/Ġ(2a),ĠwhereĠaĠisĠthe
Rollout 95, Act: 3.629
2Ġ+Ġ(ÎĶy)^2Ġ)Ġ=Ġsqrt(Ġ(4/9k2)/(245
Rollout 25, Act: 3.628
ĠformulaĠforĠanĠorbit'sĠclosestĠapproach:ĊĊr_pĠ=Ġ(h2Ġ/Ġμ)Ġ/Ġ(1Ġ+
Rollout 22, Act: 3.628
6.02214076Ġ=ĠapproxĠ0.000334
Top 16 Negative Activations
Rollout 9, Act: -6.562
.ĊĊ**FinalĠAnswer**Ċ\boxed{25}ĊĊ<|im_start|>answerĊSquareĠ\(AIME\)ĠhasĠsides
Rollout 19, Act: -6.568
}ĊĊ**FinalĠAnswer**Ċ\boxed{78}ĊĊ<|im_start|>answerĊToĠsolveĠtheĠproblem,ĠweĠneed
Rollout 2, Act: -6.574
**FinalĠAnswer**Ċ\boxed{109}ĊĊ<|im_start|>answerĊToĠfindĠtheĠremainderĠwhenĠ\(9
Rollout 13, Act: -6.591
**Ċ\boxed{187.12}ĊĊ<|im_start|>answerĊToĠdetermineĠhowĠmuchĠmoreĠmoneyĠJo
Rollout 11, Act: -6.599
ĠAnswer**Ċ\boxed{(-1,Ġ3)}ĊĊ<|im_start|>answerĊGivenĠthreeĠcomplexĠnumbersĠ\(a,
Rollout 48, Act: -6.607
.ĠTheĠchecksumĠisĠvalid.Ġ\boxed{No}ĊĊ<|im_start|>answerĊToĠdetermineĠifĠtheĠIPv4Ġheader
Rollout 66, Act: -6.613
**FinalĠAnswer**Ċ\boxed{182}ĊĊ<|im_start|>answerĊGivenĠaĠrightĠtriangleĠ\(ABC_
Rollout 81, Act: -6.619
{\dfrac{510}{11}}ĊĊ<|im_start|>answerĊGivenĠtriangleĠ\(ABC\)ĠwithĠ\
Rollout 36, Act: -6.645
**FinalĠAnswer**Ċ\boxed{192}ĊĊ<|im_start|>answerĊGivenĠaĠcubeĠwithĠedgesĠ\(\
Rollout 91, Act: -6.652
}.ĊĊ**FinalĠAnswer**Ċ\boxed{2}ĊĊ<|im_start|>answerĊGivenĠtriangleĠ\(ABC\)ĠwithĠmedian
Rollout 95, Act: -6.722
}ãĢĤĊĊ**FinalĠAnswer**Ċ\boxed{0}ĊĊ<|im_start|>answerĊTriangleĠ\(ABC\)ĠhasĠsideĠlengths
Rollout 43, Act: -6.793
ĠbeĠ\boxed{C}.ĊĊĊ**FinalĠAnswer**Ċ\boxed{C}ĊĊ<|im_start|>answerĊTheĠproblem
Rollout 43, Act: -6.817
Ġ+ĠgammaĠ|1>)Ġ\tensorproductĠ|0>ĊAnswerĠChoices:Ċ(A)ĠU_{C-NOT
Rollout 65, Act: -6.871
ĊĊ**FinalĠAnswer**Ċ\boxed{40}ĊĊ<|im_start|>answerĊGivenĠaĠrightĠrectangularĠprismĠ\(ĠP
Rollout 97, Act: -6.954
**FinalĠAnswer**Ċ\boxed{230}ĊĊ<|im_start|>answerĊGivenĠtheĠdecompositionĠreactionĠ\(2Ġ\
Rollout 21, Act: -7.162
ĠX}ĊĊ(b)Ġ\boxed{\text{Yes}}ĊĊ<|im_start|>answerĊ(a)ĠToĠshowĠthatĠCarlĠcan

DOWN_PROJ

Top 16 Positive Activations
Rollout 2, Act: 7.058
109Ġmod125.ĊĊSoĠsummaryĠsoĠfar:ĊĊPâī¡5Ġmod8ĊĊandĠP
Rollout 38, Act: 6.258
/5)Ġsqrt(x2Ġ+ĠW2)ĊĊMultiplyĠbothĠsidesĠbyĠ5:ĊĊ25(WĠ+
Rollout 18, Act: 6.139
5.ĠSinceĠxĠisĠpositive,ĠthisĠmeansĠxĠisĠaĠpositiveĠdivisorĠofĠ35.ĠTheĠdiv
Rollout 94, Act: 5.996
**FinalĠAnswer**Ċ\boxed{B}ĊĊ<|im_start|>answerĊToĠdetermineĠwhichĠstatesĠareĠentangled,Ġwe
Rollout 32, Act: 5.934
ī2,ĠwhereĠÏīĠandĠÏī2ĠareĠcomplexĠ(primitiveĠcubeĠrootsĠofĠunity).ĠSoĠÏ
Rollout 92, Act: 5.813
.78}Ġdiopters.ĊĊ<|im_start|>answerĊTheĠstudentĠisĠfarsightedĠandĠusesĠglassesĠto
Rollout 34, Act: 5.751
)Ġ+Ġexp(-2t)Ġ+ĠCĊĊMultiplyĠbothĠsidesĠbyĠexp(t):ĊĊy(t)Ġ=
Rollout 89, Act: 5.729
ĠweĠcanĠsolveĠforĠT.ĊĊSoĠfromĠhere,ĠmultiplyingĠbothĠsidesĠbyĠ5/3ĠtoĠisolateĠT
Rollout 18, Act: 5.662
ĠsoĠ3yĠâī¡1Ġmod7.ĠMultiplyĠbothĠsidesĠbyĠtheĠinverseĠofĠ3Ġmodulo7
Rollout 86, Act: 5.592
dfrac{\pi}{16}}.ĊĊ<|im_start|>answerĊToĠfindĠtheĠminimumĠvalueĠofĠtheĠintegralĠ\
Rollout 33, Act: 5.578
=3ĠcontributesĠ-48.ĠTheĠcumulativeĠtotalĠsoĠfarĠisĠ720Ġ(total)Ġ-
Rollout 89, Act: 5.568
/5)(GĠ+Ġ100)ĊĊMultiplyĠbothĠsidesĠbyĠ5:ĊĊ5GĠ=Ġ
Rollout 38, Act: 5.528
(Ġx2Ġ+ĠW2Ġ)/5ĊĊMultiplyĠbothĠsidesĠbyĠ5*6.2Ġto
Rollout 25, Act: 5.519
ĠaĠtestĠparticleĠwhenĠit'sĠlowĠmass,ĠbutĠasĠitĠbecomesĠmoreĠmassive,ĠtheĠtwo-bodyĠproblemĠbecomes
Rollout 87, Act: 5.517
ĠForĠexample,ĠsphericalĠcoordinates.ĠButĠsinceĠweĠareĠinĠaĠcube,ĠsphericalĠcoordinatesĠwonâĢĻtĠalignĠnicely.
Rollout 20, Act: 5.474
ĠtoĠunderstandĠtheĠmodulus.Ġ2021ĠdividedĠby,Ġsay,Ġ43ĠisĠ2
Top 16 Negative Activations
Rollout 96, Act: -5.568
Ġ(5Ġ-10.296,6Ġ-(-6.710))=(-5
Rollout 16, Act: -5.571
(6,3)=20,Ġ(-1)^{4}=1,ĠdenominatorĠ1Ġ-1
Rollout 91, Act: -5.583
)ĊĊVectorĠBA:ĠfromĠBĠtoĠA:Ġ(tĠ-Ġ(-1),ĠtĠ-Ġ0)Ġ=
Rollout 62, Act: -5.680
Ġby13:Ġ13*Ġ1290.88=13*1
Rollout 95, Act: -5.748
ĠABC.ĠTherefore,ĠtheĠratioĠofĠareasĠwouldĠbeĠproportionalĠtoĠtheĠsquareĠofĠscalingĠfactor.ĊĊLetĠmeĠnote
Rollout 48, Act: -5.790
x907F?ĠButĠthisĠisĠtheĠalternativeĠmethod.ĊĊWait,ĠperhapsĠthisĠisĠeasier:ĊĊWhen
Rollout 3, Act: -5.809
8,436).ĠLet'sĠsee,Ġ168^2Ġ=Ġ28,
Rollout 59, Act: -5.861
2ĠdividedĠbyĠ41?Ġ41*9Ġ=Ġ369,Ġ41
Rollout 96, Act: -5.868
9ĊĊ5tĠ-12sĠ=27Ġ+1134/169ĊĊ
Rollout 39, Act: -6.076
0Ġ+10^2Ġ+...Ġ+10^{321}ĠisĠequalĠtoĠ(1
Rollout 39, Act: -6.092
=(10Ġ+10^2Ġ+10^3)/9Ġ-Ġ(3).ĠWait,
Rollout 16, Act: -6.126
3:ĠC(6,3)*(-1)^4*[1/(1Ġ-Ġ1/8
Rollout 3, Act: -6.156
.ĠAtĠSTP,ĠtheĠmolarĠvolumeĠisĠ22.4Ġliters,ĠwhichĠisĠforĠideal
Rollout 68, Act: -6.244
4.ĊĊThenĠquadraticĠequationĠbecomesĠx2Ġ+Ġ(4Ġ-Ġ(-4))x+Ġ4=Ġx
Rollout 96, Act: -6.422
69ĊĊY_pĠ-ĠO_yĠ=27Ġ-Ġ(-1134/169
Rollout 26, Act: -6.948
ĠelementsĠinĠtheirĠstandardĠstatesĠunderĠstandardĠconditionsĠ(usuallyĠ25°CĠandĠ1Ġatm).ĊĊTheĠreaction

Layer 54

GATE_PROJ

Top 16 Positive Activations
Rollout 4, Act: 8.324
0.02Ġ/Ġ5.00Ġ=Ġ0.004.ĊĊNow,Ġsince
Rollout 59, Act: 8.167
10/28Ġ=Ġ5/14.ĊĊWait,ĠholdĠon:Ġ3+3
Rollout 72, Act: 8.141
°)ĠisĠ(sqrt(5)-1)/4?ĠWait,Ġcos(72°)Ġ=
Rollout 31, Act: 7.851
âĪļ5)/2=(1Ġ-âĪļ5)/2.ĊĊThirdly,Ġx=(1Ġ-âĪļ5
Rollout 4, Act: 7.765
327Ġ/Ġ1200ĠâīĪĠ0.003606.
Rollout 95, Act: 7.750
Ġ=Ġ(21Ġ+Ġ5âĪļ3)/2.ĠHmm,ĠseemsĠcomplicated.ĠMaybeĠcoordinateĠsystemĠis
Rollout 8, Act: 7.749
Ġ31Ġ+126=157.ĊĊButĠIĠwantĠtoĠcross-verify.ĊĊWait
Rollout 72, Act: 7.714
Ġ(sqrt(5)Ġ-Ġ1)/4Ġ*Ġ2.ĠWait,ĠmoreĠaccurately,Ġcos(
Rollout 76, Act: 7.711
Hence,ĠtheĠanswerĠisĠ929.ĊĊ**FinalĠAnswer**Ċ\boxed{929
Rollout 95, Act: 7.704
1)Ġ=Ġ(3âĪļ671)/2.ĊĊSoĠtheĠareaĠofĠtriangleĠABCĠisĠ3
Rollout 96, Act: 7.694
5.562âīĪ1.562ĊĊSimilarly,Ġy'=27Ġ-63
Rollout 92, Act: 7.686
17200Ġ/Ġ4551ĠâīĪĠ3.77938
Rollout 58, Act: 7.679
Ġ=Ġ2,019,000ĊĊ9Ġ*Ġ2019Ġ=Ġ
Rollout 59, Act: 7.664
)*(3/7)Ġ=Ġ3/28.ĊĊForĠk=1:ĠThereĠareĠthreeĠpossibilities
Rollout 84, Act: 7.645
Ġ=Ġ5Ġ+Ġ32Ġ=Ġ37.ĠWait,ĠletĠmeĠrecheckĠtheĠcalculations.ĊĊ
Rollout 99, Act: 7.644
0206)/0.75ĠâīĪĠ-0.8027467
Top 16 Negative Activations
Rollout 25, Act: -1.057
ĠrigidĠbody,ĠtheĠRocheĠlimitĠisĠapproximatelyĠdĠ=Ġ1.26Ġ*ĠR_earthĠ*
Rollout 44, Act: -1.100
,ĠbutĠifĠit'sĠcompressedĠfromĠV2ĠtoĠV1,ĠthenĠÎĶVĠ=ĠV1Ġ-ĠV
Rollout 25, Act: -1.101
Ġbody,ĠtheĠRocheĠlimitĠisĠapproximatelyĠdĠ=Ġ1.26Ġ*ĠR_earthĠ*Ġ(
Rollout 83, Act: -1.136
ĠobtainedĠfromĠtheĠotherĠbyĠrotatingĠtheĠnonagonĠinĠtheĠplane.ĠĠFindĠtheĠnumberĠofĠdistinguishableĠacceptable
Rollout 17, Act: -1.142
_0^3}ĠdVĠ=Ġ\frac{2\mu_0}{3}Ġ\math
Rollout 65, Act: -1.159
Ġ$P_{}$ĠcutsĠ$P_{}$ĠintoĠtwoĠprisms,ĠoneĠofĠwhichĠisĠsimilar
Rollout 49, Act: -1.181
Ġsubspaces,ĠsoĠXĠisĠaĠcontinuousĠimageĠofĠaĠcertainĠspace.ĊĊThisĠseemsĠtooĠvague.ĊĊAlternatively,
Rollout 25, Act: -1.219
67e-11ĊĊ=Ġ6.13e14Ġ/6.67
Rollout 99, Act: -1.222
.71145)âīĪ-0.2192Ġ-0.711
Rollout 49, Act: -1.288
,ĠtheĠspaceĠisĠcountableĠorĠaĠquotientĠofĠaĠcountableĠspace,ĠbutĠno.ĊĊAlternatively,Ġconsider
Rollout 70, Act: -1.306
.4286)ĠâīĪĠ0.3567ĊĊSo,Ġ(-1.
Rollout 49, Act: -1.404
ĠTopologyĠbyĠEngelking,ĠTheoremĠ3.8.2ĠonĠpageĠ194:
Rollout 23, Act: -1.464
)+31*ln(10)=0.6419Ġ+31*2.
Rollout 44, Act: -1.530
{p}_{1},Ġ\mathrm{~V}_{1}\right)$ĠagainstĠconstantĠpressureĠ$\mathrm{
Rollout 99, Act: -1.564
.5)/ln(10)=Ġ(-0.6931471805
Rollout 49, Act: -1.598
ĠTheoremĠ3.8.2ĠonĠpageĠ194:ĠEveryĠregularĠLindelöf

UP_PROJ

Top 16 Positive Activations
Rollout 28, Act: 6.738
2kĠ-b))Ġb2ĊĊLetĠmeĠfactorĠoutĠB:ĊĊBĠabĠ=ĠBĠ[Ġ-2
Rollout 68, Act: 6.600
Ġ+Ġ4xĠ+Ġ4.ĠRearrangingĠterms,ĠweĠget:ĊĊx^2Ġ+Ġ(
Rollout 5, Act: 6.183
))Ġ=Ġ125HĊĊLetĠmeĠfactorĠHĠout:ĊĊ(xĠ+Ġ175)
Rollout 96, Act: 6.161
xĠ-Ġ120ĊBringĠallĠtermsĠtoĠoneĠside:Ċ5xĠ-Ġ12y
Rollout 7, Act: 6.159
18NĠ=56N/3ĊĊFactorĠx:ĊĊx(AĠ-54)Ġ+1
Rollout 5, Act: 6.058
5Ġ+Ġx)Ġ=Ġ125ĊĊFactorĠtheĠsecondĠterm:ĊĊLeftĠsideĠ=Ġ(xĠ+
Rollout 45, Act: 6.042
Ġ+Ġ4bĠ+Ġ4cĊĊSimplifyĠleftĠminusĠright:ĊĊ(30Ġ-Ġ4
Rollout 96, Act: 6.040
12y)/5Ġ+5yĠ=90ĊĊMultiplyĠallĠtermsĠbyĠ5ĠtoĠeliminateĠdenominator
Rollout 84, Act: 6.014
Ġ4Ġ-Ġ4xĊĊBringĠallĠtermsĠtoĠtheĠleftĠsideĠorĠsubtractĠ4Ġ-4x
Rollout 11, Act: 6.004
Ġr3Ġ+1/r3Ġ)=0ĊĊFactorĠoutĠ2iĠsinθ:ĊĊ2iĠsin
Rollout 11, Act: 6.003
2)ĠsinθĠcosθĠ=Ġ0ĊĊFactorĠoutĠsinθ:ĊĊsinθĠ[3Ġ+Ġ(
Rollout 28, Act: 5.999
2hĠ-a)Ġ+ĠBĠkĠ=Ġ0Ġ...Ġ(1)ĊĊC(2kĠ-Ġb
Rollout 25, Act: 5.997
Ġμ_initialĠ/Ġr_p_initialĠ=Ġ0ĊĊFactorĠoutĠx:ĊĊxĠ[Ġv^4Ġ-Ġ(
Rollout 31, Act: 5.983
:ĊĊ2Ġ-Ġx^2Ġ-ĠyĠ=Ġ0ĊĊandĊĊ-Ġ(1Ġ+ĠxĠy)
Rollout 5, Act: 5.978
75Ġ+Ġx)ĊĊExpandĠtheĠleft-handĠside:ĊĊx^2Ġ-Ġ50xĠ+Ġ
Rollout 77, Act: 5.969
(x,ĠxĠ+Ġy)ĊĊSoĠifĠIĠrearrangeĠthis,ĠmaybeĠIĠcanĠexpressĠf(x,Ġx
Top 16 Negative Activations
Rollout 9, Act: -3.969
h}xĠ+Ġ10Ġ-Ġ\frac{50}{h}\).ĊĊ5.Ġ**Intersection
Rollout 96, Act: -3.974
197}Ġ=Ġ31Ċ\]ĊĊThus,ĠtheĠanswerĠis:Ċ\[Ċ\boxed
Rollout 84, Act: -3.998
ĠmĠ=Ġ5Ġ\)ĠandĠ\(ĠnĠ=Ġ32Ġ\).ĠTherefore,Ġ\(ĠmĠ+Ġn
Rollout 96, Act: -3.998
Ġ=Ġ\frac{4106Ġ\cdotĠ5Ġ+Ġ5697Ġ\cdotĠ
Rollout 66, Act: -4.036
ĠtheĠperimeterĠ\(pĠ=Ġ182\).ĊĊThus,ĠtheĠperimeterĠ\(p\)ĠisĠ\(\
Rollout 0, Act: -4.051
6}{2}Ġ=Ġ128\).ĊĊThus,ĠtheĠnumberĠofĠrationalĠnumbersĠbetweenĠ0
Rollout 66, Act: -4.058
ĠtoĠgetĠtheĠprimitiveĠtripleĠ\((84,Ġ13,Ġ85)\).ĊĊ4.
Rollout 95, Act: -4.065
3}k}{490}\)Ġrational.ĊĊThus,ĠtheĠnumberĠofĠsegmentsĠ\(\overline
Rollout 29, Act: -4.071
ĠpressureĠatĠtheĠbottomĠbeingĠ\(Ġ\rhoĠghĠ\).ĊĊ\[Ċ\boxed{A}Ċ\]<|im_end|>
Rollout 36, Act: -4.081
Ġ+Ġ\left(Ġ\frac{s^2Ġ+Ġ18864}{2s}Ġ\
Rollout 64, Act: -4.134
2012}Ġ-Ġk(2^{2009}Ġ-Ġ1)\right)
Rollout 36, Act: -4.141
Ġ+Ġ\left(Ġ\frac{s^2Ġ+Ġ8064}{2s}Ġ\right
Rollout 58, Act: -4.263
Ġ2037171Ċ\]ĊĊThus,ĠtheĠtotalĠnumberĠofĠquestionsĠonĠtheĠexam
Rollout 47, Act: -4.620
Ġ\impliesĠNĠ=Ġ9ĊĠĠĠ\]ĊĊThus,ĠtheĠvalueĠofĠ\(ĠNĠ\)Ġis
Rollout 8, Act: -4.762
mĠ=Ġ31\)ĠandĠ\(nĠ=Ġ126\).ĠThus,Ġ\(mĠ+
Rollout 59, Act: -4.770
8}ĊĠĠ\]ĊĊ-ĠForĠ\(kĠ=Ġ3\)Ġ(allĠheads):ĊĠĠ\[ĊĠ

DOWN_PROJ

Top 16 Positive Activations
Rollout 42, Act: 8.192
ĠsinceĠweĠhaveĠaĠgoodĠleavingĠgroupĠafterĠprotonationĠ(water),ĠperhapsĠE1ĠisĠpossible.ĠHowever,
Rollout 16, Act: 7.724
,953.ĊĊTherefore,ĠboxĠthat.ĊĊ**FinalĠAnswer**Ċ\boxed{7899
Rollout 3, Act: 7.285
ĪĠ28,436.ĠThenĠtakeĠsquareĠroot.ĊĊsqrt(28,43
Rollout 54, Act: 7.247
an-1-olate,ĠisĠaĠweakerĠbaseĠthanĠhydroxide?ĠNo,Ġno.ĠWait:
Rollout 98, Act: 7.196
ĠbecomeĠaĠwhiteĠdwarf,ĠitĠneedsĠtoĠgoĠthroughĠtheĠgiantĠphases.ĠButĠaĠstarĠwithĠ0.
Rollout 69, Act: 7.147
:Ġ30°Ġ/9,Ġthat'sĠ10/3°,ĠandĠsoĠon.
Rollout 54, Act: 6.950
8,ĠsoĠpropionateĠisĠaĠmuchĠweakerĠbaseĠthanĠhydroxide.ĠSoĠthatĠcorrelates:Ġhydrox
Rollout 95, Act: 6.892
ĠâĩĴĠxĠ=Ġ150/18Ġ=Ġ25/3ĠâīĪĠ8.
Rollout 11, Act: 6.832
,ĠthenĠAĠ+ĠBĠ+ĠCĠ=Ġ2ÏĢikĠforĠsomeĠintegerĠk.ĠButĠsinceĠa,
Rollout 42, Act: 6.832
atesĠtheĠhydroxylĠgroup,ĠconvertingĠitĠintoĠaĠgoodĠleavingĠgroupĠ(water).ĠThen,ĠtheĠelimination
Rollout 42, Act: 6.805
ĠgroupĠisĠprotonated,ĠformingĠaĠgoodĠleavingĠgroupĠ(water).ĠUponĠdepartureĠofĠwater,ĠaĠ**ter
Rollout 54, Act: 6.793
xideĠisĠsmallerĠandĠmayĠengageĠinĠhydrogenĠbondingĠwithĠsolventĠmore.ĠButĠmaybeĠtheĠlowerĠbasicityĠallows
Rollout 42, Act: 6.729
ylĠgroup,ĠconvertingĠitĠintoĠaĠgoodĠleavingĠgroupĠ(water).ĠThen,ĠtheĠeliminationĠwouldĠoccur,Ġremoving
Rollout 11, Act: 6.714
-realĠcĠasĠwell.ĊĊAlternatively,ĠtakeĠaĠ=Ġre^{iθ},ĠbĠ=Ġre^{iθ
Rollout 89, Act: 6.710
Ġimproper,ĠisĠtheĠanswer.ĊĊThus,Ġanswering.ĊĊ**FinalĠAnswer**Ċ\boxed{\dfrac{2
Rollout 4, Act: 6.641
.00007696ĊĊTakeĠtheĠsquareĠroot:Ġsqrt(0.00
Top 16 Negative Activations
Rollout 49, Act: -6.371
ĠitĠÏĥ-compact?ĠNo,ĠbecauseĠit'sĠnotĠsecond-countableĠandĠaĠÏĥ-compactĠspaceĠwhich
Rollout 37, Act: -6.382
5^2Ġ=Ġ9*125Ġ-Ġ27*25Ġ=Ġ11
Rollout 5, Act: -6.419
Ġbase.ĠTheĠsegmentĠthatĠjoinsĠtheĠmidpointsĠofĠtheĠlegsĠdividesĠtheĠtrapezoidĠintoĠtwoĠregions
Rollout 48, Act: -6.432
ĠJustĠaĠidentifierĠforĠfragmentedĠpackets.ĊĊThenĠcomeĠFlagsĠandĠFragmentĠOffset:Ġ"4000".ĠLet
Rollout 27, Act: -6.449
26Ġ+10)ĠâīĪĠ3.318e-16ĊĊThenĠdivide
Rollout 70, Act: -6.474
ĠinĠbase-7:ĠSinceĠ7^3=343,Ġ363-3
Rollout 44, Act: -6.502
ĠtoĠmatchĠtheĠinternalĠpressureĠatĠeveryĠstep,ĠhenceĠmaximumĠworkĠisĠdone.ĠWait,Ġno:Ġwait,
Rollout 70, Act: -6.544
Ġ(>333),ĠsoĠ7^3=343Ġ(>333,ĠsoĠ
Rollout 70, Act: -6.693
7ĠofĠ210:ĊĊ7^3=343>210.ĠSoĠ
Rollout 19, Act: -6.744
ĠaboutĠn=3.ĠThenĠsumĠcubes1+8+27=36.ĠDividedĠby
Rollout 98, Act: -7.021
)^3ĠâīĪĠ10^10Ġ/27ĠâīĪĠ370ĠmillionĠyears
Rollout 44, Act: -7.185
ĠexpansionĠintoĠvacuum,ĠnoĠexternalĠpressure.ĠTherefore,ĠnoĠworkĠdoneĠ(W=0).ĠAlso,Ġsince
Rollout 70, Act: -7.919
0Ġ/7^3=2400/343âīĪ6.997,
Rollout 37, Act: -8.423
^2Ġ=Ġ9*125Ġ-Ġ27*25Ġ=Ġ112
Rollout 37, Act: -8.486
=2,Ġk2=3:Ġ4*27Ġ-8*9=108
Rollout 70, Act: -8.866
10Ġ+d0Ġ=2*(d3*343Ġ+d2*49Ġ+

Layer 55

GATE_PROJ

Top 16 Positive Activations
Rollout 22, Act: 3.456
/s.ĠButĠinĠseveralĠdecimalĠplaces.ĊĊĊ**FinalĠAnswer**ĊTheĠmeanĠmolecularĠspeedĠofĠhydrogenĠgasĠ(H
Rollout 4, Act: 3.359
ĠisĠapproximatelyĠ0.88%.ĊĊ**FinalĠAnswer**ĊTheĠpercentageĠerrorĠinĠtheĠcalculationĠofĠtheĠenergy
Rollout 26, Act: 3.234
:Ċ\[Ċ\boxed{90.4}Ċ\]<|im_end|>
Rollout 4, Act: 3.204
[Ċ\boxed{0.88\%}Ċ\]<|im_end|>
Rollout 22, Act: 3.204
Ġ\text{m/s}Ġ\)ĊĊ**FinalĠAnswer**:Ċ\[Ċ\boxed{169
Rollout 23, Act: 3.183
Ġ\,Ġ\text{min}^{-1}ĊĠĠĠĠĠ\]ĊĠĠĠ-ĠBetaĠdecayĠhalf-lifeĠ\(ĠT
Rollout 42, Act: 3.173
ĠgroupĠisĠprotonated,ĠformingĠaĠgoodĠleavingĠgroupĠ(water).ĠUponĠdepartureĠofĠwater,ĠaĠ**ter
Rollout 3, Act: 3.163
69}Ġ\,Ġ\text{m/s}Ċ\]<|im_end|>
Rollout 91, Act: 3.160
Ġ(tĠ-Ġ1,Ġt)\)ĠandĠ\(\overrightarrow{CB}Ġ=Ġ(-2,
Rollout 27, Act: 3.155
Ġ\text{K}ĊĠĠĠĠĠ\]ĊĊ**FinalĠAnswer:**ĊTheĠrmsĠspeedĠofĠtheĠBeĠnucleiĠis
Rollout 15, Act: 3.132
ĠofĠlampsĠ\(ĠnĠ\)ĠisĠaĠpowerĠofĠ2.ĠThisĠresultĠisĠderivedĠfromĠpropertiesĠofĠlinear
Rollout 91, Act: 3.081
ĠtoĠfindĠtheĠvalueĠofĠ\(|\cotĠBĠ-Ġ\cotĠC|\).ĊĊ1.Ġ**CoordinateĠSetup**
Rollout 33, Act: 3.077
ĠwithĠ\(1Ġ\leqĠkĠ\leqĠ5\),ĠatĠleastĠoneĠofĠtheĠfirstĠ\(
Rollout 99, Act: 3.075
Ġapproximately2.1573.ĊĊ**FinalĠAnswer**ĊTheĠdecimalĠvalueĠofĠ\(Ġx_3Ġ\
Rollout 23, Act: 3.065
)}{0.098}Ġ\approxĠ7.0724Ġ\,Ġ\text
Rollout 92, Act: 3.050
3.78Ġdiopters.ĊĊ**FinalĠAnswer**ĊTheĠrefractiveĠpowerĠofĠtheĠglassesĠlensesĠis
Top 16 Negative Activations
Rollout 51, Act: -8.126
1/7Ġ]ĊĊWhichĠis:ĊĊ[1/9Ġ-1/7]Ġ-Ġ[Ġ-1/
Rollout 93, Act: -8.141
ĠnotĠanĠintegral.ĠHmmm.ĠĊĊWait,ĠanotherĠthought:ĠwhenĠdealingĠwithĠsumsĠoverĠrootsĠofĠunity
Rollout 30, Act: -8.145
a_{n+1}(a_{n+1}Ġ-1)Ġb_nĠ)ĊĊTherefore,Ġ
Rollout 30, Act: -8.165
Ġa_{n+1}(a_{n+1}Ġ-1Ġ)Ġ]ĊĊHmm.ĠSoĠnow,
Rollout 44, Act: -8.166
ĠworkĠdoneĠonĠtheĠgasĠisĠp1*(V2Ġ-ĠV1)ĠifĠtheĠexternalĠpressureĠisĠp
Rollout 12, Act: -8.171
)Ġ/Ġsqrt(Ġ(Ġ(t/ν)^2Ġ-1Ġ)(1Ġ-Ġt^2)Ġ)ĊĊ
Rollout 51, Act: -8.183
7ĊĊSoĠtheĠintegralĠis:ĊĊ(1/9Ġ-1/7)Ġ-Ġ(Ġ-1/
Rollout 30, Act: -8.195
^âĪŀĠb_nĠ(1/a_{n+1}Ġ-Ġ1/a_{n+3}}
Rollout 12, Act: -8.240
Ġ(Ġ(1/ν)Ġsqrt(Ġ(t2Ġ-Ġν2)(1Ġ-Ġt2)Ġ))
Rollout 51, Act: -8.241
ĠtheĠsecondĠbracketĠfromĠtheĠfirst:ĊĊ(1/9Ġ-1/7)Ġ-Ġ(1/9
Rollout 30, Act: -8.244
ĠtermĠisĠb_nĠ(1/a_{n+1}Ġ-Ġ1/a_{n+3})
Rollout 51, Act: -8.346
Ġ+Ġ1/7Ġ]ĊĊ=Ġ(1/9Ġ-Ġ1/7)Ġ-Ġ(-1/
Rollout 12, Act: -8.360
Ġ(t2Ġ-Ġν2)ĠbecomesĠ(ν2Ġ-Ġν2)Ġ=0,ĠsoĠthatĠfirst
Rollout 30, Act: -8.420
a_{n+1}(a_{n+1}Ġ-1Ġ)Ġ=Ġ(a_{n-
Rollout 30, Act: -8.458
a_{n+1}(a_{n+1}Ġ-1)Ġb_nĠ)Ġ=Ġ[Ġ(
Rollout 51, Act: -8.815
/7ĊĊTherefore:ĊĊResultĠ=Ġ(1/9Ġ-Ġ1/7)Ġ-Ġ(-1/

UP_PROJ

Top 16 Positive Activations
Rollout 20, Act: 4.019
}\)ĠforĠallĠpositiveĠintegersĠ\(ĠaĠ\).ĊĊForĠ\(\sigma(a^n)Ġ\equivĠ1
Rollout 32, Act: 3.867
Ġ+Ġ0Ġ+Ġ1Ġ=Ġ2\)ĊĊTotalĠsumĠforĠoneĠcycle:Ġ\(2(1
Rollout 35, Act: 3.806
4)R(n)Ġ\)ĠwhereĠ\(ĠR(n)Ġ\)ĠisĠanotherĠpolynomialĠwithĠintegerĠcoefficients.
Rollout 91, Act: 3.805
)\).ĊĊ3.Ġ**Calculations**:ĊĠĠĠ-Ġ**AngleĠ\(B\)**:ĊĠĠĠĠĠ-Ġ\
Rollout 56, Act: 3.803
3}Ġ\),ĠandĠ\(ĠA_1A_2Ġ=Ġ2Ġ\).ĠTheĠareaĠofĠthis
Rollout 16, Act: 3.796
^k}}Ċ\]ĊĊCalculatingĠeachĠtermĠseparately:ĊĊ-ĠForĠ\(Ġk=1Ġ\):Ġ\(Ġ\
Rollout 96, Act: 3.793
94}{2197}Ċ\]ĊĊFinally,ĠsummingĠ\(\alpha\)ĠandĠ\(\
Rollout 31, Act: 3.760
Ġ+Ġxy)Ġa_mĠ=Ġ0Ċ\]ĊSinceĠthisĠmustĠholdĠforĠallĠ\(Ġa_0
Rollout 44, Act: 3.733
Ġ\DeltaĠUĠ\)ĠisĠnegative,ĠnotĠpositive.ĠStatementĠ[D]ĠisĠincorrect.ĊĊFinalĠAnswer:
Rollout 44, Act: 3.722
ĠpressureĠresultsĠinĠmoreĠworkĠdoneĠonĠtheĠgas.ĠTherefore,ĠstatementĠ[A]ĠisĠcorrect.ĊĊ**[
Rollout 95, Act: 3.716
3}k}{490}\).ĊĠĠĠ-ĠForĠthisĠlengthĠtoĠbeĠrational,Ġ\(\sqrt
Rollout 83, Act: 3.712
rotationsĠbyĠ0ĠtoĠ8Ġpositions).ĠHowever,ĠonlyĠtheĠidentityĠrotationĠ(0Ġpositions)Ġfixes
Rollout 40, Act: 3.698
ĠgreatestĠintegerĠlessĠthanĠorĠequalĠtoĠ\(|p|Ġ+Ġ|q|\).ĊĊUsingĠcomplexĠnumbers,Ġthe
Rollout 59, Act: 3.697
ĠnumberĠofĠheadsĠ\(k\)ĠareĠcalculatedĠasĠfollows:ĊĊ-ĠForĠ\(kĠ=Ġ0\)Ġ(all
Rollout 95, Act: 3.686
sqrt{3}k\)ĠmustĠbeĠrational.ĠHowever,Ġ\(\sqrt{3}\)ĠisĠirrational
Rollout 21, Act: 3.658
.ĠIfĠ\(ĠfĠ\)ĠisĠtheĠfloorĠfunction,ĠthenĠ\(ĠxĠ-Ġ\lfloorĠxĠ\rf
Top 16 Negative Activations
Rollout 89, Act: -6.478
ĠweĠgiveĠinĠtermsĠofĠ66.666...,ĠbutĠifĠbeingĠinĠprimaryĠschool,Ġthe
Rollout 73, Act: -6.499
98.ĠTherefore,ĠanswerĠisĠ398.ĊĊButĠlet'sĠconfirmĠwithĠanotherĠsmallĠcase.
Rollout 8, Act: -6.510
0ĠdifferentĠcards,ĠhenceĠtheĠnumberĠofĠorderedĠsequencesĠisĠ10P5.ĠIfĠtheyĠareĠind
Rollout 89, Act: -6.512
0ĠdividedĠbyĠ3,ĠsoĠaboutĠ33.333...ĠsoĠ2/5
Rollout 89, Act: -6.516
Ġ3,ĠsoĠunits=33.33.ĠAlternatively,Ġgirls=Ġ2*33
Rollout 8, Act: -6.561
5ĠcardsĠinĠorderĠwouldĠbeĠ10P5,ĠasĠearlier.ĊĊButĠtheĠproblemĠhereĠisĠthat
Rollout 89, Act: -6.600
100Ġ/0.6=166,666...ĊĊButĠgirlsĠareĠ0
Rollout 89, Act: -6.605
5)/3ĠâīĪ166.666.ĠButĠhowĠcanĠthatĠbe?ĊĊWait,
Rollout 89, Act: -6.698
,Ġgirls=Ġ2*33.33=66.666.ĊĊHmm,
Rollout 8, Act: -6.703
ĠtheĠnumberĠofĠorderedĠsequencesĠisĠ10P5.ĠIfĠtheyĠareĠindistinct,ĠthenĠthe
Rollout 89, Act: -6.752
3.ĠAlternatively,Ġgirls=Ġ2*33.33=66.666
Rollout 89, Act: -6.907
0ĠdividedĠbyĠ3,ĠsoĠunits=33.33.ĠAlternatively,Ġgirls=Ġ2
Rollout 76, Act: -6.911
ĠFromĠfourĠsubgrids,ĠtheĠnumberĠofĠpairsĠisĠC(4,2)=6.ĠButĠnot
Rollout 89, Act: -6.924
ĠpartĠisĠ100/3âīĪ33.333...ĠTherefore,Ġ2Ġparts
Rollout 89, Act: -7.027
ĠofĠgirlsĠwouldĠbeĠapproximatelyĠ66.67.ĠButĠpractically,Ġit'sĠaroundĠ67
Rollout 95, Act: -7.073
25/3ĠâīĪĠ8.333...ĊĊThenĠxĠ=Ġ25/3

DOWN_PROJ

Top 16 Positive Activations
Rollout 22, Act: 8.616
2016Ġkg/mol.ĠThen,ĠdivideĠbyĠAvogadro'sĠnumber:Ġ0.0
Rollout 80, Act: 7.647
11/(86400)ĠâīĪ1.157e6ĠdaysĠâīĪ
Rollout 71, Act: 7.311
,ĠtheĠsumĠofĠtheĠproductsĠofĠtheĠrootsĠtakenĠtwoĠatĠaĠtime:ĠabĠ+ĠbcĠ+ĠacĠ=
Rollout 22, Act: 7.188
.016e-3Ġkg/mol)Ġ/Ġ6.0221407
Rollout 54, Act: 7.015
ĠisĠaĠstrongerĠacid),ĠsoĠit'sĠtheĠleastĠnucleophilicĠamongĠtheĠchargedĠspecies.ĠMethanolĠ(
Rollout 54, Act: 6.995
ĠaproticĠsolvents,ĠalkoxideĠisĠmoreĠnucleophilic,ĠbutĠinĠpolarĠprotic,Ġdue
Rollout 75, Act: 6.984
ĠquickĠcheck,ĠRamanujanĠprovedĠthatĠtheĠonlyĠnumbersĠrequiringĠsixĠpositiveĠsquaresĠareĠ1,Ġ
Rollout 22, Act: 6.971
02016Ġkg/mol.ĠDividedĠbyĠAvogadro'sĠnumberĠ(6.02
Rollout 40, Act: 6.935
Ġ+1Ġ)Ġ]Ġ*Ġ(1Ġ+Ġi)ĊĊWait,Ġno.ĠWait,ĠtheĠtermĠwasĠ1
Rollout 22, Act: 6.883
3Ġkg/mol)Ġ/Ġ6.02214076e23ĠmolâĪĴ
Rollout 22, Act: 6.807
Ġ0.002016Ġ/Ġ6.02214076
Rollout 54, Act: 6.790
Ġactually,ĠstericĠhindranceĠmightĠmakeĠitĠlessĠnucleophilicĠbecauseĠtheĠloneĠpairsĠareĠharderĠtoĠaccess
Rollout 37, Act: 6.783
3Ġ-Ġ3^3*5^2Ġ=Ġ9*125Ġ-Ġ27
Rollout 54, Act: 6.757
Ġbe:ĊĊ5.ĠEthanethiolateĠ(mostĠnucleophilic)ĊĊ2.ĠHydroxideĊĊ1
Rollout 57, Act: 6.754
ĠUĠaroundĠxĠwithĠclosureĠcontainedĠinĠA.ĠButĠholdĠon,ĠinĠtheĠdensityĠtopology,ĠclosuresĠmightĠnot
Rollout 22, Act: 6.730
.022e23ĠisĠ0.002016Ġ/Ġ6.
Top 16 Negative Activations
Rollout 87, Act: -6.075
))ĠâĪ«Ġ(ÏģĠ/Ġr)ĠdV,ĠwhereĠÏģĠisĠtheĠchargeĠdensity,ĠrĠis
Rollout 62, Act: -6.079
ĠbyĠ121.ĊĊ121*143=Ġ17303.
Rollout 56, Act: -6.097
.112Ġ+...Ġconverges.ĊĊIfĠweĠassumeĠeachĠdifferenceĠisĠmultipliedĠbyĠ1/2
Rollout 38, Act: -6.133
Ġ)/31=Ġ(42kĠ+888kĠ)/31=93
Rollout 25, Act: -6.179
,ĠthenĠtechnically,ĠtheĠgravitationalĠparameterĠwouldĠbeĠμĠ=ĠG*(M_earthĠ+ĠM_asteroid
Rollout 56, Act: -6.209
Ġ/12Ġ)Ġ=Ġ7âĪļ3Ġ/144ĊĊSoĠtotal:ĊĊ=Ġ|Ġ0
Rollout 8, Act: -6.248
4/3024=744÷24=31;Ġ302
Rollout 73, Act: -6.293
ĠremainderĠisĠ398.ĠHowever,ĠIĠshouldĠcheckĠtheĠanswerĠmoreĠcarefully.ĠWait,ĠbutĠbefore
Rollout 76, Act: -6.324
29.ĊĊButĠholdĠon,ĠbeforeĠfinalizing,ĠcheckĠtheĠarithmeticĠonceĠmore.ĊĊTotalĠcoloringsĠwith
Rollout 48, Act: -6.429
ĊĊ5Ġ+6=ĠBĊĊ4Ġ+ĠBĠ=ĠFĠ(hex).ĠSoĠforĠtheĠsum,Ġit
Rollout 49, Act: -6.504
Alternatively,ĠuseĠtheĠfactĠthatĠXĠhasĠaĠcountableĠextentĠ(everyĠclosedĠdiscreteĠsubspaceĠisĠcountable
Rollout 59, Act: -6.516
92=Ġ515.ĊĊButĠlet'sĠcheck,ĠsomeĠotherĠway.ĠMaybeĠthinkĠofĠgenerating
Rollout 62, Act: -6.602
7=Ġ7*297=2079.Ġ2081-2
Rollout 53, Act: -6.654
ĠthatĠinĠaĠtriangle,ĠtheĠcentroidĠdividesĠeachĠmedianĠintoĠaĠratioĠofĠ2:1,ĠwithĠthe
Rollout 87, Act: -6.690
Ġ+Ġz2)]ĠdxĊĊToĠcomputeĠthisĠintegral,ĠIĠcanĠuseĠsubstitution.ĠLetâĢĻsĠsetĠuĠ=
Rollout 53, Act: -7.101
iansĠintersectĠatĠcentroidĠP,ĠwhichĠdividesĠeachĠmedianĠintoĠratioĠ2:1.ĊĊGivenĠthatĠPDĠ=

Layer 56

GATE_PROJ

Top 16 Positive Activations
Rollout 51, Act: 9.910
Ġ(0Ġ-Ġ0)/nĠ=Ġ0ĊĊTherefore,Ġcos7xĠandĠcos9xĠintegrated
Rollout 31, Act: 9.664
,ĠandĠy=(1+âĪļ5)/2.ĊĊ**FinalĠAnswer**ĊTheĠpossibleĠvaluesĠofĠ\(Ġy
Rollout 17, Act: 9.600
3Ġ[Ġ2/3Ġ]Ġ-Ġ2ĊĊ=Ġ2Ġ-Ġ2Ġ=Ġ0ĊĊ
Rollout 76, Act: 9.521
Hence,ĠtheĠanswerĠisĠ929.ĊĊ**FinalĠAnswer**Ċ\boxed{929
Rollout 59, Act: 9.505
Ġ*7^2.ĠNoĠcommonĠfactors.ĠTherefore,ĠmĠ=123,Ġn=39
Rollout 16, Act: 9.361
104Ġ-Ġ112)/7Ġ=Ġ(-8)/7ĠâīĪ-1.1
Rollout 61, Act: 9.360
Ġ10Ġâī¡Ġ1ĠmodĠ9ĊĊSoĠtheĠmultiplicativeĠorderĠofĠ2ĠmodĠ
Rollout 13, Act: 9.352
ĠtoĠannualĠisĠ$187.12.ĊĊ**FinalĠAnswer**Ċ\boxed{187
Rollout 59, Act: 9.347
Ġ=123,Ġn=392.ĊĊSoĠ123Ġ+Ġ392
Rollout 61, Act: 9.336
Ġcorrect,ĠleadingĠtoĠtheĠanswerĠ913.ĊĊ**FinalĠAnswer**Ċ\boxed{913
Rollout 90, Act: 9.297
ĠtheĠfirstĠkĠdigits,ĠMĠisĠtheĠmiddleĠdigitĠ(0Ġâī¤ĠMĠâī¤Ġ9),ĠandĠLĠis
Rollout 78, Act: 9.293
Ġ49.ĠTheĠanswerĠisĠ49.ĊĊ**FinalĠAnswer**Ċ\boxed{49}ĊĊ
Rollout 62, Act: 9.277
121ĠisĠ11ÃĹ11,ĠsoĠtwoĠprimeĠfactors.ĠOkay,ĠsoĠnowĠwe
Rollout 32, Act: 9.266
ĠofĠtheĠentriesĠisĠ4018.ĠSoĠthat'sĠprobablyĠtheĠanswer.ĊĊWait,Ġlet's
Rollout 39, Act: 9.264
ĠmeĠcheck.ĊĊ10^{322}ĠisĠ1ĠfollowedĠbyĠ322Ġzeros.
Rollout 69, Act: 9.261
12Ġ+192=204.ĊĊSo,Ġyes,ĠconclusionĠisĠa=12
Top 16 Negative Activations
Rollout 33, Act: -4.882
'sĠk)ĠinsertingĠtheĠnumbersĠappropriately.ĠButĠperhapsĠnot.ĊĊAlternatively,ĠmaybeĠIĠcanĠuseĠtheĠprincipleĠof
Rollout 95, Act: -4.893
???ĠWait.ĊĊWait,ĠholdĠon.ĠLetĠmeĠrecastĠthis.ĊĊPointĠPkĠisĠalongĠAB.
Rollout 24, Act: -4.913
ĠperhapsĠusingĠextremalĠprinciplesĠorĠinequalities?ĊĊButĠnotĠsure.ĊĊAlternatively,ĠthinkĠofĠRĠasĠselectorsĠforĠthe
Rollout 64, Act: -4.951
ĠexpandingĠtheĠcoefficientĠofĠa1:ĊĊButĠperhapsĠnotĠhelpful.ĊĊAlternatively,ĠdivideĠbothĠsidesĠbyĠ2^{
Rollout 69, Act: -4.983
Ġtan(3Ġtheta)Ġexpression,ĠbutĠthatĠmightĠnotĠhelpĠhere.ĊĊAlternatively,ĠsinceĠtheĠproblemĠinvolvesĠan
Rollout 75, Act: -5.003
ĠmayĠcomplicateĠthings.ĠWait,Ġno,ĠperhapsĠnot.ĊĊAlternatively,ĠnoticeĠthatĠ8NĠ+3
Rollout 57, Act: -5.016
ĠwithĠclosureĠinĠtheĠdensityĠtopology.ĠButĠthisĠseemsĠdifficult.ĊĊAlternatively,ĠtakeĠtwoĠnon-emptyĠclosedĠsetsĠin
Rollout 93, Act: -5.073
,ĠwhichĠcomplicatesĠmatter.ĊĊAlternatively,ĠitĠmightĠnotĠbeĠusefulĠdirectlyĠbecauseĠtheĠrelationsĠdependĠonĠparityĠof
Rollout 49, Act: -5.124
ĠplusĠmetrizableĠimpliesĠsecondĠcountable?ĠMaybeĠnotĠtheĠdirectionĠhere.ĊĊWait,ĠbutĠourĠspaceĠX
Rollout 1, Act: -5.156
ĠorthocentricĠsystemĠorĠsomethingĠsimilar?ĊĊLetĠmeĠthink.ĊĊSupposeĠweĠfixĠy.ĠThenĠtheĠconditions
Rollout 66, Act: -5.159
ĠtoĠassignĠaĠandĠbĠparityĠyet.ĠLet'sĠproceed.ĊĊSo,ĠstartingĠwithĠabĠ=Ġ6(a
Rollout 1, Act: -5.186
2)}Ġ/Ġ2.ĊĊButĠperhapsĠit'sĠhard.ĊĊWait,ĠbackĠtoĠtheĠequation:ĊĊForĠy
Rollout 91, Act: -5.218
ĠHmm.ĠLet'sĠthink.ĊĊButĠperhapsĠit'sĠbetterĠtoĠassignĠcoordinates.ĠLetĠmeĠplaceĠtheĠtriangleĠin
Rollout 49, Act: -5.245
Ġsomething?ĊĊButĠIâĢĻmĠnotĠsure.ĠLetĠmeĠthink.ĠIfĠeveryĠsubsetĠofĠXĠisĠFÏĥ,
Rollout 14, Act: -5.420
ĠIĠtoĠFĠandĠK.ĊĊNo,ĠthatĠseemsĠnotĠimmediatelyĠhelpful.ĠĊĊWait,ĠhaveĠyouĠconsiderĠ
Rollout 64, Act: -5.445
,ĠexpandingĠtheĠcoefficientĠofĠa1:ĊĊButĠperhapsĠnotĠhelpful.ĊĊAlternatively,ĠdivideĠbothĠsidesĠbyĠ2

UP_PROJ

Top 16 Positive Activations
Rollout 28, Act: 8.933
Ġ(3)^2*(3)^2=0Ġ?ĊĊWaitĠno,ĠwithĠa=3,Ġb=
Rollout 67, Act: 8.687
Ġ3936Ġ+1020.ĊĊCalculate:ĊĊ-72*64=Ġ-
Rollout 96, Act: 8.454
20Ġ-4106=ĠsomethingĠwrong.ĊĊWait,Ġno:ĊĊWait,Ġ410
Rollout 96, Act: 8.428
9Ġ+1134)/169.ĊĊWait,ĠinĠpreviousĠcalculations,ĠOB_xĠwasĠ(-
Rollout 0, Act: 8.307
Ġ2,3,5,7ĠareĠeven?ĊĊWaitĠfloorĠoperations:ĊĊForĠprimeĠ2:Ġ
Rollout 59, Act: 8.296
Ġ16*24.5?ĠWait,Ġ16Ġ*24.5ĠisĠ
Rollout 36, Act: 8.278
688,ĠSá»ijĠnÃłyĠdÃłiĠquá.ĠWait,Ġ18,864Ġsquared:ĠLet
Rollout 53, Act: 8.239
1971ĠdividedĠbyĠ146:ĊĊ1971Ġ÷146:ĊĊ
Rollout 70, Act: 8.234
51ĠisĠaĠ7-10Ġdouble.ĊĊ51ĠinĠbase-7ĠisĠ10
Rollout 96, Act: 8.182
7+6+9+3=34?ĊĊWait,Ġwait:ĊĊ5+8=13
Rollout 36, Act: 8.120
5,688,ĠSá»ijĠnÃłyĠdÃłiĠquá.ĠWait,Ġ18,864Ġsquared
Rollout 2, Act: 8.103
999*10Ġ+9,Ġwait,ĠperhapsĠ9999=9*1
Rollout 71, Act: 8.076
92+492Ġ+102.ĠLetĠmeĠcomputeĠthat:ĊĊ392=Ġ
Rollout 96, Act: 8.011
740Ġ/Ġ169.ĊĊWait,ĠletĠmeĠcomputeĠthat.ĠDivideĠnumeratorĠandĠdenominatorĠby
Rollout 59, Act: 8.005
4.5ĠisĠ392?ĠNo,Ġ16Ġ*24ĠisĠ38
Rollout 22, Act: 7.991
ĠbyĠ0.006333.ĊĊComputeĠ18,168.7
Top 16 Negative Activations
Rollout 69, Act: -5.554
10^\circ}{3^i}\right)Ġ-Ġ\sec\left(\frac{30
Rollout 28, Act: -5.569
Ġ(-kĠa)/(b(2hĠ-a))Ġ-ĠhĠb/(a(2kĠ-b
Rollout 29, Act: -5.570
ĠtheĠhandleĠisĠplugged,ĠpreventingĠflow,ĠthenĠthisĠcan'tĠhappen.ĠTherefore,ĠhowĠdoesĠthisĠaffectĠthe
Rollout 12, Act: -5.571
K}{dk}Ġ=Ġ\frac{E(k)Ġ-Ġ(1Ġ-Ġk^2)K(k
Rollout 69, Act: -5.573
Ġ\left(Ġ\sec(10^\circ)Ġ-Ġ1Ġ\right).Ċ\]ĊĊHowever,
Rollout 24, Act: -5.637
ĠpairĠ(R,ĠC),ĠtheĠsetĠRĠhasĠtoĠbeĠaĠminimalĠcoveringĠsetĠforĠtheĠrowsĠwithĠrespect
Rollout 24, Act: -5.750
ĠR.ĠInĠaĠminimalĠpair,ĠtheseĠpartialĠordersĠmustĠhaveĠcertainĠpropertiesĠthatĠfixĠtheirĠsizes.ĊĊAlternatively,
Rollout 14, Act: -5.843
sqrt{J}}{2\sqrt{A}}Ġ-Ġ\frac{K}{A^2}
Rollout 28, Act: -5.918
2ĠkĠa2Ġ/Ġ(2hĠ-a)Ġ-Ġ2ĠhĠb2Ġ/Ġ(2k
Rollout 69, Act: -5.954
Ġ\left(Ġ\sec(10^\circ)Ġ-Ġ\lim_{iĠ\toĠ\infty
Rollout 87, Act: -5.960
3Ġ\ln(1Ġ+Ġ\sqrt{2})Ġ\right]Ġ}ĊĊSinceĠIĠrecallĠthisĠfromĠinternet
Rollout 69, Act: -6.008
(Ġ\frac{2}{\sqrt{3}}Ġ-Ġ1Ġ\right)Ġ=Ġ\frac{
Rollout 69, Act: -6.153
}{4}Ġ\left(Ġ\sec(\theta)Ġ-Ġ\sec(3\theta)Ġ\right
Rollout 91, Act: -6.221
|Ġ\frac{tĠ+Ġ1}{t}Ġ-Ġ\frac{1Ġ-Ġt}{t}
Rollout 30, Act: -6.234
frac{b_n}{a_{n+1}}Ġ-Ġ\frac{1}{a_{n+
Rollout 66, Act: -6.539
:ĊĠĠĠ\[ĊĠĠĠ\frac{ab}{cĠ-Ġa}Ġ=Ġ6pĊĠĠĠ\

DOWN_PROJ

Top 16 Positive Activations
Rollout 17, Act: 10.382
Ġlet'sĠthink.ĠInĠelectromagnetism,ĠtheĠmagneticĠfieldĠofĠaĠdipoleĠcanĠbeĠwrittenĠas:ĊĊ
Rollout 40, Act: 8.877
.ĠSinceĠ36ÏĢĠisĠjustĠ18ĠfullĠcircles,ĠequivalentĠtoĠ0.ĠSoĠe
Rollout 17, Act: 8.773
First,ĠIĠneedĠtoĠrecallĠtheĠformulaĠforĠtheĠmagneticĠfieldĠofĠaĠdipole.ĠTheĠmagneticĠfieldĠ**
Rollout 6, Act: 8.755
ĠwithĠaĠsparseĠoracle.Ġ(ThisĠisĠtheĠMeyer,ĠIĠthink,Ġtheorem.)ĠSoĠperhapsĠthisĠisĠthe
Rollout 68, Act: 8.753
ence,ĠtheĠdesiredĠanswerĠis501.ĊĊ**FinalĠAnswer**Ċ\boxed{501}ĊĊ
Rollout 4, Act: 8.753
ĠpercentageĠerrorĠisĠapproximatelyĠ0.88%.ĊĊ**FinalĠAnswer**ĊTheĠpercentageĠerrorĠinĠtheĠcalculationĠof
Rollout 17, Act: 8.511
ĠforĠtheĠmagneticĠfieldĠofĠaĠdipole.ĠTheĠmagneticĠfieldĠ**B**(r)ĠproducedĠbyĠaĠdip
Rollout 17, Act: 8.442
Ġ\timesĠ10^{-2}}ĠT.ĊĊ**FinalĠAnswer**Ċ\boxed{2.54
Rollout 8, Act: 8.261
Ġ+126=Ġ157.ĊĊ**FinalĠAnswer**Ċ\boxed{157}ĊĊ
Rollout 61, Act: 8.206
,ĠleadingĠtoĠtheĠanswerĠ913.ĊĊ**FinalĠAnswer**Ċ\boxed{913}ĊĊ
Rollout 47, Act: 8.110
ĠtheĠanswerĠNĠ=Ġ9.ĠLooksĠgood.ĊĊ**FinalĠAnswer**Ċ\boxed{9}ĊĊ<|im_start|>answer
Rollout 42, Act: 8.110
ĠhydrogenĠfromĠtheĠcyclopropaneĠ(whichĠisĠhighlyĠstrained)ĠcouldĠtransferĠtoĠtheĠcarbocation.ĠBut
Rollout 27, Act: 8.060
8.0ÃĹ10^6ĠK.ĊĊ**FinalĠAnswer**ĊTheĠrmsĠspeedĠofĠtheĠBeĠnuclei
Rollout 76, Act: 7.969
ence,ĠtheĠanswerĠisĠ929.ĊĊ**FinalĠAnswer**Ċ\boxed{929}ĊĊ
Rollout 59, Act: 7.965
HenceĠtheĠanswerĠisĠ515.ĊĊ**FinalĠAnswer**Ċ\boxed{515}ĊĊ
Rollout 0, Act: 7.932
2^{8-1}=128.ĊĊ**FinalĠAnswer**Ċ\boxed{128}ĊĊ
Top 16 Negative Activations
Rollout 97, Act: -8.187
ĠstoichiometricĠcoefficient.ĠHmm,Ġno.ĠMaybeĠstep-by-step.ĊĊLetĠmeĠmodelĠtheĠpartialĠpressuresĠas
Rollout 16, Act: -8.233
73...ĊĊNow,ĠaddingĠupĠallĠtheseĠtermsĠtermĠbyĠterm:ĊĊStartĠwithĠ12Ġ(from
Rollout 27, Act: -8.237
ĠisĠconfusing.ĠWait,Ġlet'sĠtryĠtoĠmoveĠstepĠbyĠstep.ĠTheĠproblemĠfirstĠasksĠtoĠcalculateĠV
Rollout 39, Act: -8.369
ĠmoreĠpreciseĠwayĠis:ĊĊLet'sĠperformĠtheĠdivisionĠstepĠbyĠstep.ĠConsiderĠthatĠ999...
Rollout 67, Act: -8.394
=348Ġ*a*d.ĊĊWait,ĠstepĠbyĠstep:ĊĊMultiplyĠequationĠ3ĠbyĠa*d
Rollout 38, Act: -8.427
ĠthisĠisĠgettingĠaĠbitĠconfusing.ĠLet'sĠapproachĠstepĠbyĠstep.ĊĊFirst,ĠdefineĠcoordinateĠsystem:ĠLet
Rollout 16, Act: -8.682
/2)^k).ĠLet'sĠcomputeĠthisĠagainĠtermĠbyĠterm.ĊĊk=1:ĊĊC(6
Rollout 53, Act: -8.800
ĠwhereĠcalculationsĠcanĠbeĠperformed.ĠLet'sĠtryĠthisĠstepĠbyĠstep.ĊĊLetĠmeĠsetĠcoordinates:ĊĊLetâĢĻs
Rollout 28, Act: -8.929
2*(3n)^2=0ĊĊĠcomputeĠterm-wise:ĊĊ4*9n2h2=
Rollout 17, Act: -9.129
29e-11)^3Ġ)ĊĊComputeĠstepĠbyĠstep:ĊĊNumerator:2*4*
Rollout 83, Act: -9.212
ĠconsideringĠsymmetries.ĊĊButĠletĠmeĠtakeĠitĠstepĠbyĠstep.ĊĊFirst,ĠhowĠmanyĠassignmentsĠinĠtotal
Rollout 45, Act: -10.250
Ġ(-3b)Ġ+0Ġ+4.ĊĊComputeĠtermĠbyĠterm:ĊĊ25Ġ+6aĠ+2
Rollout 11, Act: -10.263
1/(a2ĠzĠ)*ĠaĊĊSimplifyĠtermĠbyĠterm:ĊĊ-ĠaĊ-ĠaĠzĊ
Rollout 51, Act: -10.426
Ġ+Ġ1/7Ġ]Ġ=Ġlet'sĠcomputeĠtermĠbyĠterm.ĊĊFirstĠtermĠinsideĠfirstĠbrackets:Ġ
Rollout 19, Act: -10.529
ĠisĠ0.ĊĊWait,ĠletĠmeĠdoĠthisĠstepĠbyĠstep.ĠLet'sĠcomputeĠ\(Ġ(kĠ-
Rollout 50, Act: -10.587
0}101]Ġ\]ĊĊSimplifyĠtermĠbyĠterm:ĊĊ-Ġ\((\log_{1

Layer 57

GATE_PROJ

Top 16 Positive Activations
Rollout 12, Act: 9.226
,ĠandĠwhenĠxĠ=1,Ġt=Ġν.ĊĊButĠthenĠxĠ=Ġt/ν,ĠdxĠ=
Rollout 12, Act: 9.057
Ġ-Ġt2)Ġ)Ġ/ĠtĠ]ĠdθĊĊTherefore,ĠdtĠ/Ġ[Ġsqrt(Ġ(t2
Rollout 12, Act: 8.931
t2)Ġ)Ġ]Ġ=ĠdθĠ/ĠtĊĊSo,Ġrearranged,ĠtheĠintegrandĠdtĠ/
Rollout 73, Act: 8.858
ĠsoĠforĠiĠfromĠ0ĠtoĠj-1.ĠSoĠforĠeachĠj,Ġ2^jĠappears
Rollout 56, Act: 8.800
ĠinĠtheĠtriangle,ĠweĠneedĠtoĠfigureĠoutĠcoordinates.ĊĊWait,Ġactually,ĠstartingĠwithĠaĠright-angled
Rollout 69, Act: 8.784
3Ġ-3âĪļ3Ġ)Ġ/12âĪļ3Ġ)Ġ=Ġ(6Ġ-3âĪļ3Ġ)/(
Rollout 12, Act: 8.782
nuĠ/k^2Ġ)Ġ/Ġ(Ġnu2Ġ)ĊĊ=Ġ[ĠE(k)Ġ-Ġnu2ĠK(k
Rollout 96, Act: 8.773
ĠandĠ2148=-12k.ĊĊFromĠfirstĠequation,Ġk=-895/
Rollout 51, Act: 8.719
ĠdxĠ=Ġ[-cos(nx)/n]0^pi=Ġ(-cos(nĠpi)/nĠ+Ġcos0
Rollout 73, Act: 8.708
:ĠsumĠshouldĠbeĠ1*2Ġ=2.ĊĊPlugĠinĠn=1:Ġ(1Ġ-1
Rollout 30, Act: 8.680
_{n+1}Ġ-1Ġ)Ġb_n.ĊĊSo,ĊĊb_nĠ=Ġa_{n+3
Rollout 87, Act: 8.675
1Ġ+Ġr2),ĠtĠdrĠ=ĠrĠdt.ĊĊWait,ĠpreviousĠsubstitution.ĠSinceĠtĠ=Ġsqrt(
Rollout 17, Act: 8.608
/T,ĠandĠÏĢĠa03ĠisĠm3.ĊĊCalculatingĠunits:ĊĊ(ĠT·m/AĠ*
Rollout 51, Act: 8.583
0Ġ+Ġsin8θ*(-4/63)Ġ=Ġ-4/63Ġsin8θ.ĊĊ
Rollout 17, Act: 8.577
Ġμ_pĠ)Ġ/Ġ(3ĠÏĢĠa03Ġ)ĊĊSoĠinĠnumbers:ĊĊ2*(4ÏĢe-
Rollout 8, Act: 8.570
,k)Ġ*ĠP(5,5Ġ-k)]ĊĊWait,ĠforĠk=1:ĊĊNumberĠforĠeach
Top 16 Negative Activations
Rollout 21, Act: -4.839
ProposedĠbyĠCarlĠSchildkraut*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠpartĠ(
Rollout 7, Act: -4.843
ĠofĠnon-promotedĠĠ$47$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠI'm
Rollout 94, Act: -4.845
,dĊ(D)Ġa,bĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠdetermineĠwhich
Rollout 54, Act: -4.846
3,ĠBĠ=Ġ2Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtackleĠthisĠquestionĠstep
Rollout 43, Act: -4.848
imes\left|0\right\rangleĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtry
Rollout 16, Act: -4.850
Ġ.ĊĊ*ProposedĠbyĠYannickĠYao*Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠEmilia
Rollout 97, Act: -4.861
ĠofĠ$\mathrm{Y}$?Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠsee.ĠIĠhave
Rollout 27, Act: -4.898
ĠofĠvelocityĠalongĠtheĠlineĠofĠsight).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠI'veĠgotĠthis
Rollout 3, Act: -4.944
ĠradonĠ(Rn)ĠinĠm/sĊ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠIĠneedĠto
Rollout 33, Act: -4.965
ĠpermutationĠisĠgreaterĠthanĠ$k$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠfind
Rollout 1, Act: -4.966
Ġ$Ġ\mathcal{H}$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠsoĠIĠneedĠtoĠshow
Rollout 32, Act: -4.969
M}^{2009}.$Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠsoĠIĠneed
Rollout 24, Act: -4.991
ĠcontainĠtheĠsameĠnumberĠofĠrows.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠstartĠbyĠtryingĠto
Rollout 17, Act: -5.017
}|$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>thinkĊAlright,ĠletĠmeĠtryĠtoĠtackleĠthis
Rollout 26, Act: -5.036
~mol}^{-1}$Ġ]Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,Ġlet'sĠtryĠtoĠfigureĠthis
Rollout 66, Act: -5.070
$Ġ.ĠFindĠ$p$Ġ.Ċ<|im_start|>assistantĊ<|im_start|>thinkĊOkay,ĠletĠmeĠtryĠtoĠworkĠthrough

UP_PROJ

Top 16 Positive Activations
Rollout 9, Act: 8.191
ĠtoĠmakeĠthisĠclearer.ĊĊLet'sĠplaceĠtheĠsquareĠonĠaĠcoordinateĠsystem.ĠLet'sĠsayĠpointĠAĠis
Rollout 91, Act: 8.014
ĠbetterĠtoĠassignĠcoordinates.ĠLetĠmeĠplaceĠtheĠtriangleĠinĠcoordinateĠsystem.ĠLetĠmeĠsetĠpointĠBĠatĠ(-
Rollout 95, Act: 7.818
Ġit'sĠaĠrightĠtriangle?ĠLetĠmeĠcheckĠusingĠtheĠPythagoreanĠtheorem.ĠIfĠABCĠisĠaĠright
Rollout 38, Act: 7.803
ĠbetweenĠsouthwestĠandĠsoutheastĠcorners.ĊĊWait,Ġlet'sĠbuildĠaĠcoordinateĠsystem.ĠLetâĢĻsĠconsiderĠtheĠnortheastĠcorner
Rollout 9, Act: 7.759
ĠonĠtheĠleftĠsideĠofĠEM.ĠWait,ĠinĠourĠcoordinateĠsystem,ĠEMĠisĠatĠx=10
Rollout 9, Act: 7.665
ĠmakeĠthisĠclearer.ĊĊLet'sĠplaceĠtheĠsquareĠonĠaĠcoordinateĠsystem.ĠLet'sĠsayĠpointĠAĠisĠat
Rollout 2, Act: 7.643
?ĠmodĠ1000ĊĊSolveĠthisĠsystemĠusingĠChineseĠRemainderĠTheorem.ĊĊWeĠneed
Rollout 11, Act: 7.610
ĠmustĠhaveĠnon-realĠcoefficients,ĠmakingĠitĠhardĠtoĠapplyĠthat.ĊĊAlternatively,ĠthinkĠofĠlogĠvariables?ĠIf
Rollout 38, Act: 7.532
Ġ+336=0ĊĊNow,ĠweĠhaveĠquadraticĠinĠr:ĊĊ336Ġr2Ġ-
Rollout 53, Act: 7.525
.ĊĊAlternatively,Ġit'sĠpossibleĠtoĠplaceĠthisĠtriangleĠintoĠcoordinatesĠwhereĠcalculationsĠcanĠbeĠperformed.ĠLet'sĠtry
Rollout 59, Act: 7.437
Ġ515.ĊĊButĠlet'sĠcheck,ĠsomeĠotherĠway.ĠMaybeĠthinkĠofĠgeneratingĠfunctions?ĊĊThe
Rollout 58, Act: 7.405
171}.ĊĊButĠletĠmeĠseeĠifĠthatĠmakesĠsense.ĠLetĠmeĠdoĠaĠsmallerĠcase.
Rollout 81, Act: 7.402
1Ġ=Ġ0ĊĊLet'sĠattemptĠtoĠsolveĠthisĠquadraticĠequationĠforĠh.ĠQuadraticĠformula:Ġh
Rollout 81, Act: 7.369
3,ĠDC=17.ĠThenĠinĠsuchĠaĠcase,Ġsometimes,ĠinĠcertainĠtrianglesĠwithĠgivenĠratios
Rollout 29, Act: 7.367
ĠaĠhigherĠpressureĠatĠtheĠbottom.ĊĊWait,ĠthisĠseemsĠcontradictory.ĠLet'sĠmakeĠitĠclearer.ĠIfĠthe
Rollout 37, Act: 7.366
mondeĠmatricesĠlookĠlikeĠ[[1,Ġx,Ġx^2,Ġ...,Ġx^{n-1}],
Top 16 Negative Activations
Rollout 75, Act: -4.444
Ġ42Ġ+Ġ32.ĠThatĠworks.ĠSimilarly,Ġ32=9:Ġ9
Rollout 93, Act: -4.459
.ĠHenceĠsum=1007ĠasĠwell.ĠWait,ĠjĠcanĠgoĠupĠto201
Rollout 71, Act: -4.529
ĠSoĠdâīĪ25ĠorĠdâīĪ26.ĠButĠdĠcanĠbeĠupĠtoĠ25Ġsince
Rollout 71, Act: -4.581
ĠbyĠprimesĠupĠtoĠsqrt(2011).Ġsqrt(2011)ĠisĠapproxĠ
Rollout 38, Act: -4.627
6)ĠisĠlessĠthanĠ1055.ĠComputeĠ10542=(10
Rollout 90, Act: -4.632
Ġ12108ĠisĠ5Ġdigits.ĠTheĠmiddleĠdigitĠisĠtheĠthirdĠone,ĠwhichĠis
Rollout 3, Act: -4.684
Ġroot.ĊĊsqrt(28,436).ĠLet'sĠsee,Ġ168^2
Rollout 3, Act: -4.712
^2Ġ=Ġ28,224.Ġ169^2Ġ=Ġ28
Rollout 22, Act: -4.712
Ġ2,890,000.ĠTherefore,Ġsqrt(2,869,
Rollout 51, Act: -4.748
Ġ[sinnĠpiĠ-Ġsinn0]/n.ĠButĠsinceĠnĠisĠinteger,ĠsinnĠpiĠis
Rollout 19, Act: -4.896
,ĠitsĠdivisorsĠareĠ1Ġand83.ĠHence,ĠonlyĠdivisorsĠâī¥6Ġare83
Rollout 19, Act: -4.900
ĠofĠ83ĠgreaterĠthanĠorĠequalĠto6.ĠSinceĠ83ĠisĠprime,ĠitsĠdivisors
Rollout 90, Act: -4.916
ĠwouldĠresultĠinĠBĠbeingĠ1090.ĠIsĠ1090ĠdivisibleĠbyĠ2
Rollout 71, Act: -4.935
4-300=7744.Ġ7744=882.
Rollout 72, Act: -4.998
),ĠtheĠonlyĠrootĠofĠunityĠisĠ1Ġitself.ĠSoĠweĠneedĠtoĠaddĠthreeĠ1sĠto
Rollout 71, Act: -5.107
1Ġ-1875=136.Ġ136ĠisĠnotĠaĠperfectĠsquare.ĊĊ

DOWN_PROJ

Top 16 Positive Activations
Rollout 25, Act: 10.921
ĠâīĪĠ6.8e-5Ġm/s2.ĠEvenĠsuchĠaĠmassiveĠasteroidĠ(1e
Rollout 49, Act: 10.312
arilyĠFÏĥ"ĠorĠsomething?ĊĊButĠIâĢĻmĠnotĠsure.ĠLetĠmeĠthink.ĠIfĠeveryĠsubsetĠof
Rollout 49, Act: 10.311
ixsonĠdecompositionĠisĠcountable.ĠButĠIĠamĠnotĠsure.ĊĊAlternatively,ĠthinkĠaboutĠtheĠfactĠthatĠif
Rollout 89, Act: 10.262
.ĠIfĠtheyĠuseĠfractionsĠinĠboys.ĠMaybeĠtheyĠjustĠrequireĠanĠexactĠanswerĠevenĠifĠnotĠaĠwholeĠnumber
Rollout 28, Act: 10.215
ĠknownĠasĠtheĠnine-pointĠellipse.ĠButĠI'mĠnotĠexactlyĠsure.ĠItĠrequiresĠmoreĠknowledge.ĊĊInĠany
Rollout 16, Act: 10.177
Ġtree.ĠEachĠsaplingĠindependentlyĠsucceedsĠwithĠprobabilityĠ1/2ĠeachĠday.ĠSoĠtheĠprocessĠcontinuesĠuntil
Rollout 93, Act: 10.085
ĠanĠintegralĠoverĠsomeĠfunction,ĠbutĠagainĠI'mĠnotĠsure.ĊĊWait,Ġlet'sĠtryĠaĠsmallĠcase
Rollout 28, Act: 9.845
ĠfeetĠofĠaltitudesĠandĠcentroids...ĠbutĠI'mĠnotĠsure.ĊĊAlternatively,ĠseeĠthatĠtheĠlocusĠisĠknown
Rollout 28, Act: 9.841
ĠconicsĠareĠhyperbolasĠbecauseĠtheĠdiscriminantĠB2Ġ-4AC>0ĠforĠallĠnon-de
Rollout 49, Act: 9.834
ĠOrĠmaybeĠhasĠcountableĠextent?ĠButĠI'mĠnotĠsure.ĊĊAlternatively,ĠinĠaĠhereditarilyĠLind
Rollout 49, Act: 9.664
ĠandĠHausdorffĠimpliesĠregular?ĠI'mĠnotĠsure.ĊĊWait,Ġactually,ĠaĠhereditarily
Rollout 49, Act: 9.658
Ġright-separatedĠspacesĠareĠcountable).ĠButĠI'mĠnotĠsure.ĊĊHere'sĠaĠtheorem:ĠeveryĠLindel
Rollout 27, Act: 9.580
ĠrmsĠfluctuationĠinĠenergyĠwouldĠbeĠsqrt(⣨E2⣩Ġ-Ġ⣨E⣩2).ĠLet's
Rollout 5, Act: 9.532
Ġareas,ĠandĠthenĠfindĠtheĠgreatestĠintegerĠnotĠexceedingĠx2/100.ĠĊĊFirst,Ġlet
Rollout 22, Act: 9.477
ĠdividedĠbyĠkg/molĠgivesĠ(kg·m2/s2)/kg)Ġ=Ġm2/s2.
Rollout 71, Act: 9.392
ĠaddĠupĠtoĠ-c,ĠandĠtheirĠproductĠisĠc2Ġ-Ġ2011.ĠTherefore,
Top 16 Negative Activations
Rollout 49, Act: -8.443
,ĠhenceĠBorel.ĠButĠBernsteinĠsetsĠareĠnotĠBorel.ĠTherefore,ĠsuchĠaĠBernsteinĠset
Rollout 49, Act: -8.451
ff,Ġuncountable),ĠthereĠareĠsubsetsĠthatĠareĠnotĠFÏĥ.ĠForĠexample,ĠtheĠsetĠof
Rollout 49, Act: -8.463
ĠuncountableĠclosedĠset.ĠSuchĠaĠsetĠisĠnotĠBorel,ĠhenceĠnotĠFÏĥ.ĠBut
Rollout 65, Act: -8.490
1995*q.ĠSinceĠpĠandĠqĠareĠcoprime,ĠthenĠpĠdivides199
Rollout 49, Act: -8.491
ĠeveryĠsubsetĠisĠFÏĥ,ĠthenĠeveryĠsubsetĠisĠaĠcountableĠunionĠofĠclosedĠsets.ĠTherefore,Ġthe
Rollout 16, Act: -8.569
ĠbeĠinĠtheĠformĠm/nĠwhereĠmĠandĠnĠareĠcoprime,ĠandĠthenĠcomputeĠ10
Rollout 49, Act: -8.572
,ĠinĠcompleteĠmetricĠspaces,ĠtheĠirrationalsĠareĠaĠGδĠset,ĠbutĠcomplementĠofĠGδ
Rollout 49, Act: -8.701
ĠeveryĠsubsetĠisĠFÏĥ,ĠthenĠeveryĠsubsetĠisĠaĠcountableĠunionĠofĠclosedĠsets,ĠeachĠofĠwhich
Rollout 49, Act: -8.786
ĠisĠFÏĥ,ĠthenĠeveryĠsubsetĠisĠaĠcountableĠunionĠofĠclosedĠsets.ĠTherefore,ĠtheĠBore
Rollout 36, Act: -8.826
92}.ĊĊButĠoriginalĠproblemĠseemsĠtoĠoriginateĠfromĠaĠcompetitionĠlikelyĠanswerĠisĠinteger,ĠwhichĠ1
Rollout 54, Act: -8.832
Ġbasic,ĠhydroxideĠisĠsmallerĠandĠmayĠengageĠinĠhydrogenĠbondingĠwithĠsolventĠmore.ĠButĠmaybeĠtheĠlower
Rollout 49, Act: -8.979
.ĠForĠexample,ĠtheĠsetĠofĠirrationalsĠisĠnotĠFÏĥ.ĠWait,Ġactually,ĠinĠcomplete
Rollout 57, Act: -8.984
ĠbeĠseparated.ĠAsĠaĠclassicalĠexample,ĠconsiderĠtwoĠdisjointĠLebesgueĠmeasureĠzeroĠsets.ĠThereĠexistĠtwo
Rollout 49, Act: -9.320
,ĠconsiderĠaĠBernsteinĠset.ĠAĠBernsteinĠsetĠisĠuncountable,ĠmeetsĠeveryĠperfectĠset,ĠsoĠit
Rollout 49, Act: -9.407
ĠCantorĠsetĠareĠnotĠFÏĥ,ĠlikeĠtheĠco-FÏĥĠsets.ĠForĠexample,ĠaĠGδ
Rollout 57, Act: -10.533
ĠAsĠaĠclassicalĠexample,ĠconsiderĠtwoĠdisjointĠLebesgueĠmeasureĠzeroĠsets.ĠThereĠexistĠtwoĠdenseĠmeasureĠzero

Layer 58

GATE_PROJ

Top 16 Positive Activations
Rollout 13, Act: 3.569
[Ċ1.12^2Ġ=Ġ1.2544Ġ\quadĠ\text{and
Rollout 42, Act: 3.065
.ĠTheĠhydroxylĠgroupĠisĠprotonated,ĠformingĠaĠgoodĠleavingĠgroupĠ(water).ĠUponĠdepartureĠof
Rollout 13, Act: 2.835
Ċ1.12^2Ġ=Ġ1.12Ġ*Ġ1.12.ĠLet
Rollout 42, Act: 2.752
ĠtypicallyĠundergoĠE1ĠmechanismsĠbecauseĠtheyĠcanĠformĠmoreĠstableĠcarbocations.ĊĊSoĠtheĠinitialĠstepĠwouldĠbeĠproton
Rollout 6, Act: 2.711
ĠMeyer'sĠtheoremĠequivalenceĠbetweenĠ\(Ġ\textbf{P}/\text{poly}Ġ\)ĠandĠsparse
Rollout 13, Act: 2.708
ĠâīĪĠ6000Ġ*Ġ1.57351936ĠâīĪ
Rollout 42, Act: 2.706
ĠaqueousĠH3PO4,ĠwhichĠisĠaĠgoodĠacid,Ġit'sĠmoreĠlikelyĠanĠE1Ġmechanism
Rollout 42, Act: 2.696
ĠgroupĠisĠprotonated,ĠformingĠaĠgoodĠleavingĠgroupĠ(water).ĠUponĠdepartureĠofĠwater,ĠaĠ**ter
Rollout 42, Act: 2.691
ĠintoĠaĠgoodĠleavingĠgroupĠ(water).ĠThen,ĠtheĠeliminationĠwouldĠoccur,ĠremovingĠaĠprotonĠ(H+)
Rollout 42, Act: 2.674
O+),ĠitĠleavesĠasĠwater,ĠgeneratingĠaĠcarbocation.ĠWait,ĠbutĠifĠtheĠmechanismĠisĠE
Rollout 42, Act: 2.647
utane-derived)Ġring.ĠĊĊUltimately,ĠeliminationĠofĠaĠbeta-hydrogenĠresultsĠinĠtheĠformationĠof
Rollout 23, Act: 2.607
ĠofĠtheĠ115ĠBqĠgives3.17%ÃĹ115âīĪ3.
Rollout 42, Act: 2.607
ane-derived)Ġring.ĠĊĊUltimately,ĠeliminationĠofĠaĠbeta-hydrogenĠresultsĠinĠtheĠformationĠofĠa
Rollout 54, Act: 2.567
.ĊĊRegardingĠpartĠB:ĊĊWeĠhaveĠtoĠchooseĠstatementĠ1ĠorĠ2.ĊĊStatements:ĊĊ1.ĠIn
Rollout 42, Act: 2.454
ĠtheĠcyclopropaneĠcanĠactĠasĠaĠconjugatedĠdiradicalĠandĠcombineĠwithĠtheĠcarbocation.
Rollout 20, Act: 2.447
Ġmodulo43.ĠSince2ĠhasĠmultiplicativeĠorderĠ14Ġmod43Ġ(asĠÏĨ(4
Top 16 Negative Activations
Rollout 71, Act: -8.369
ĠrootsĠofĠtheĠquadraticĠx2Ġ+ĠcxĠ+Ġ(c2Ġ-Ġ2011).ĠBecauseĠif
Rollout 64, Act: -8.378
+36Ġ?ĠNotĠsure.ĊĊAlternatively,ĠtheĠanswerĠisĠ2^{2}*3^{2}(
Rollout 11, Act: -8.388
ĠtoĠtheĠsumĠofĠtheĠvariablesĠplusĠtheĠsumĠofĠtheirĠproductsĠtwoĠatĠaĠtime.ĠThatĠformĠmakesĠme
Rollout 17, Act: -8.410
/s2)Ġ)Ġ/ĠT?ĠWait,ĠmaybeĠit'sĠsimplerĠtoĠnoteĠthatĠunitsĠcheckĠout:ĊĊ**
Rollout 28, Act: -8.462
2/a2).ĠSoĠtheĠcoefficientsĠare:ĊĊForĠx2:Ġa2/b2ĊĊForĠxy:
Rollout 69, Act: -8.479
1}ĊĊAlternatively,Ġlet'sĠsupposeĠweĠhaveĊĊLetâĢĻsĠdefineĠsomeĠrecurrenceĠhere.ĠSupposeĠweĠsetĠx
Rollout 59, Act: -8.531
ĠsquaresĠofĠeachĠprobability:ĊĊ(3/28)2Ġ+Ġ(10/28)2
Rollout 75, Act: -8.568
ĠallĠdistinctĠandĠodd.ĠWeĠneedĠtoĠshowĠthatĠa2Ġ+Ġb2Ġ+Ġc2ĠcanĠbeĠwritten
Rollout 16, Act: -8.568
3.ĊĊWait,ĠbutĠthisĠseemsĠlarge.ĠTheĠanswerĠisĠ789,953?
Rollout 71, Act: -8.585
ĠaddĠupĠtoĠ-c,ĠandĠtheirĠproductĠisĠc2Ġ-Ġ2011.ĠTherefore,
Rollout 51, Act: -8.633
'sĠcomputeĠIntegralĠAĠandĠIntegralĠB.ĊĊSo,Ġlet'sĠcomputeĠIntegralĠA:ĠâĪ«02ÏĢ
Rollout 16, Act: -8.771
2,Ġ...,ĠT6}.ĠSoĠtheĠexpectedĠvalueĠweĠneedĠisĠE[T].ĊĊSo,ĠweĠcanĠuse
Rollout 69, Act: -9.125
ĠaboutĠmanipulatingĠtheĠtermĠspecifically.ĠLet'sĠwrite:ĊĊLetâĢĻsĠlookĠatĠtheĠtermĠterm_iĠ=Ġsin2
Rollout 69, Act: -9.214
ĠusingĠtheĠtangentĠfunction.ĊĊWait,ĠanotherĠidea.ĠLetĠmeĠnoteĠthatĠ3Ġtheta_iĠ=Ġtheta_{
Rollout 95, Act: -9.248
ĠifĠtheĠrightĠangleĠisĠatĠB,ĠbutĠ92Ġ+Ġ(5âĪļ3)2Ġ=Ġ
Rollout 69, Act: -9.285
,ĠletĠmeĠthinkĠaboutĠmanipulatingĠtheĠtermĠspecifically.ĠLet'sĠwrite:ĊĊLetâĢĻsĠlookĠatĠtheĠtermĠterm

UP_PROJ

Top 16 Positive Activations
Rollout 64, Act: 5.039
*(2^{2009}Ġ-1Ġ)Ġ+8Ġ-1=8*(2^{2
Rollout 7, Act: 4.710
56NĠ-56P=56NĠ+15P=66NĊĊSo
Rollout 18, Act: 4.646
Ġ-5Ġ-25k)/3=Ġ(0Ġ-9k)/3=Ġ-3k.ĊĊ
Rollout 62, Act: 4.559
Ġ-mĠ+11Ġâī¡m2Ġ-mĠ+1Ġâī¡0Ġmod5.ĊĊSoĠm
Rollout 11, Act: 4.515
5/4)Ġe^{i2θ}Ġ-Ġ(15/4)Ġe^{-i2
Rollout 81, Act: 4.473
*400Ġ+2601*84=1,040,40
Rollout 75, Act: 4.428
4Ġ+Ġ4Ġ+Ġ1Ġ+Ġ9Ġ+Ġ1Ġ=Ġ35\),ĠwhichĠuses
Rollout 28, Act: 4.356
ĠhkĠ+Ġb2Ġh2Ġ+a2Ġk2Ġ-ĠaĠb2ĠhĠ-Ġa2bĠk
Rollout 67, Act: 4.349
Ġ2ĠdĠqĠ)Ġ]ĠnĠ+Ġ[ĠaĠpĠ-ĠaĠqĠ-ĠpĠdĠ+ĠdĠqĠ]ĊĊ
Rollout 81, Act: 4.321
/3Ġ-Ġh/17Ġ)/(Ġ1Ġ+Ġ(h/17)(-h/
Rollout 2, Act: 4.312
999=999*10Ġ+9,Ġwait,ĠperhapsĠ9999
Rollout 90, Act: 4.297
2018*(300Ġ+80Ġ+8)=2018*30
Rollout 64, Act: 4.245
1Ġ]Ġ/Ġ(2^{2009}Ġ-1Ġ)=8Ġ+7/(2^{
Rollout 62, Act: 4.241
Ġ-1Ġ+11=121k2Ġ+11kĠ+11=11
Rollout 28, Act: 4.178
)Ġk2Ġ-4bĠhĠ-4aĠkĠ+ab=0ĊĊAssumingĠaĠandĠb
Rollout 28, Act: 4.170
Ġ(-kĠa)/(b(2hĠ-a))Ġ-ĠhĠb/(a(2kĠ-b
Top 16 Negative Activations
Rollout 99, Act: -7.610
ĠandĠ"what'sĠtheĠdecimalĠvalueĠofĠx3".ĊĊWait,ĠbutĠstartingĠfromĠx0,Ġx1
Rollout 72, Act: -7.613
ĠbutĠorderĠn,ĠwhichĠusuallyĠmeansĠaĠpositiveĠinteger.ĊĊTherefore,ĠtheĠanswerĠisĠallĠevenĠpositiveĠintegers.
Rollout 58, Act: -7.625
37,171.ĠThat'sĠcorrect.ĊĊButĠweĠneedĠtoĠmakeĠsureĠthatĠtheĠresultĠis
Rollout 43, Act: -7.646
ĠtoĠ|1⣩ĠwhenĠcontrolĠisĠ|0⣩.ĊĊTherefore,ĠtheĠoutputĠisĠ(ĠalphaĠ|0⣩
Rollout 15, Act: -7.651
ĠallĠlampsĠwillĠbeĠoffĠafterĠsomeĠtime.ĊĊWait,ĠbutĠIĠneedĠtoĠbeĠcarefulĠhere.ĠTheĠproblem
Rollout 51, Act: -7.662
ĠHowever,Ġsin8θĠstillĠisĠaĠrealĠnumber.ĊĊButĠmaybeĠweĠcanĠwriteĠabsoluteĠvalue?ĠWait,
Rollout 19, Act: -7.697
ĠTherefore,Ġonly.Therefore,Ġn=78.ĊĊButĠinĠtheĠproblemĠstatement,ĠinĠtheĠoriginalĠmodulus
Rollout 88, Act: -7.723
ĠchannelsĠintoĠaĠvector,ĠreferringĠtoĠweights.ĠHmm.ĊĊWait,ĠletĠmeĠcheckĠagain:Ġ"WhatĠis
Rollout 99, Act: -7.734
1.ĠWhat'sĠtheĠdecimalĠvalueĠofĠx3?"ĊĊWait,ĠeachĠiterationĠusesĠtheĠlastĠthreeĠpoints.
Rollout 59, Act: -7.743
ĠgetĠsquaredĠasĠ(3/28)^2?ĊĊWait,ĠbutĠwait,ĠsoĠoriginallyĠeachĠofĠtheir
Rollout 44, Act: -7.744
ĠeveryĠstep,ĠhenceĠmaximumĠworkĠisĠdone.ĠWait,Ġno:Ġwait,ĠinĠcompression,ĠtheĠworkĠdone
Rollout 76, Act: -7.745
,ĠtheĠsquaresĠinĠtheirĠunionĠmustĠallĠbeĠred.ĊĊYes,ĠbecauseĠforĠS1ĠtoĠbeĠallĠred
Rollout 55, Act: -7.777
/2],ĠwithĠtheĠmiddleĠentryĠasĠ1.ĊĊLetĠmeĠwriteĠthatĠout:ĊĊ\[Ċ\math
Rollout 43, Act: -7.869
.ĠSoĠyes,Ġit'sĠcontrolĠfollowedĠbyĠtarget.ĊĊThus,ĠanswerĠCĠisĠcorrect.ĠThenĠtheĠmistake
Rollout 43, Act: -8.049
ĠcontrolĠ(firstĠqubit)ĠisĠ|0⣩.ĊĊSoĠfirstĠqubitĠisĠcontrol,ĠsecondĠisĠtarget
Rollout 73, Act: -8.129
0ĠtermĠwhichĠisĠ0,ĠsoĠsameĠresult.ĊĊButĠperhapsĠIĠshouldĠconfirm.ĊĊAlternatively,ĠIĠrecall

DOWN_PROJ

Top 16 Positive Activations
Rollout 92, Act: 13.999
/61.5âīĪ0.016260ĊĊSubtract:Ġ0.
Rollout 25, Act: 13.421
12.714e6âīĪ6.85e14ĊĊThisĠisĠapproximationĠfor
Rollout 56, Act: 13.280
1/1008ĠâīĪ1.827ĊĊThenĠ*ĠâĪļ3âīĪ1
Rollout 25, Act: 13.279
1.2714e7âīĪ6.849e14Ġm^3/s
Rollout 92, Act: 13.062
.4054)Ġ-Ġ(1.6260)Ġ=Ġ3.77
Rollout 53, Act: 13.043
=414/73ĠâīĪ5.67.ĊĊSimilarly,ĠsumĠoverĠx_iĠy
Rollout 92, Act: 12.904
.615)ĠisĠapproximatelyĠ-1.62601626.ĠSoĠadding
Rollout 49, Act: 12.775
InĠgeneral,ĠeveryĠsubsetĠofĠaĠcompactĠmetricĠspaceĠisĠÏĥ-compact?ĠNo.ĠTheĠspaceĠ{0
Rollout 3, Act: 12.328
98)Ġ)ĠâīĪĠsqrt(Ġ~225,753Ġ)ĠâīĪĠ4
Rollout 99, Act: 11.950
6315)ĠâīĪĠ1.1573ĊĊThusĠx3ĠâīĪĠ
Rollout 99, Act: 11.932
486715)âīĪ1.15733466671
Rollout 49, Act: 11.805
Alternatively,ĠconsiderĠthatĠinĠX,ĠeveryĠsubspaceĠisĠÏĥ-compact,ĠsoĠinĠparticular,ĠXĠitself
Rollout 92, Act: 11.208
54054Ġ-0.01626Ġ=Ġ0.0378
Rollout 99, Act: 11.190
4563)ĠâīĪĠ1.1575ĊĊTherefore,Ġx3Ġ=Ġx
Rollout 99, Act: 11.169
(-3.4563)=1.1573.ĊĊThus,Ġx3=x2
Rollout 3, Act: 11.127
Ġ/0.697433âīĪ26,034.21.ĊĊsqrt
Top 16 Negative Activations
Rollout 62, Act: -7.941
ĠisĠcalledĠaĠsphenicĠnumberĠifĠit'sĠsquare-freeĠwithĠthreeĠdistinctĠprimes,ĠbutĠhereĠitĠcan
Rollout 25, Act: -7.980
Ġnegative.ĠHence,Ġμ_finalĠisĠnegative,ĠwhichĠisĠimpossible.ĠThatĠsuggestsĠthatĠgivenĠtheseĠparametersĠ(original
Rollout 95, Act: -8.016
ĠconclusionĠafterĠdetailedĠcalculation.ĠTherefore,ĠtheĠanswerĠisĠ\boxed{0}ãĢĤĊĊ**FinalĠAnswer**Ċ\
Rollout 26, Act: -8.020
083Ġ*Ġ298ĠâīĪĠ0.0083Ġ*Ġ30
Rollout 55, Act: -8.064
Ġ÷Ġ12Ġ=Ġ168.166...,ĠsoĠ168*
Rollout 3, Act: -8.088
ĠbutĠasĠperĠmyĠknowledgeĠcutoffĠisĠ2023,ĠsoĠifĠneedĠtoĠanswer,ĠinĠthe
Rollout 66, Act: -8.198
.ĠTherefore,ĠtheĠaltitudeĠisĠ(leg1Ġ*Ġleg2)/hypotenuse.ĊĊGivenĠthatĠABC0
Rollout 33, Act: -8.205
6*6=36ĊĊk=4:24*2=48ĊĊk=
Rollout 36, Act: -8.306
0ĊĊThenĠcomputeĠ8642Ġ=Ġ746,496ĊĊAndĠcross
Rollout 82, Act: -8.327
9âīĪ1.6279*0.623âīĪ1.015.ĊĊ
Rollout 36, Act: -8.327
Ġ+Ġ1,728s2Ġ+Ġ746,496ĊĊSimilarly,
Rollout 86, Act: -8.390
/8.ĠNumericalĠvalue:Ġpi^3âīĪ31.006,ĠsoĠ3
Rollout 53, Act: -8.402
)=729/292=ĠapproxĠ2.496Ġapproximately.ĊĊTherefore,Ġgiven
Rollout 96, Act: -8.733
down:5000*895=4,475,000ĊĊ
Rollout 71, Act: -8.804
,Ġc2Ġ>2011Ġ=>Ġc>=45Ġ(sinceĠ442=1
Rollout 36, Act: -9.572
37728s2Ġ+Ġ355690496)/(4s

Layer 59

GATE_PROJ

Top 16 Positive Activations
Rollout 20, Act: 10.294
Ġ+7Ġ+23Ġ+43Ġ+47.ĠLetĠmeĠaddĠthat:ĊĊ2Ġ+Ġ3
Rollout 61, Act: 9.642
5Ġ+19+13Ġ+7Ġ+1=ĠindeedĠlet'sĠcompute:ĊĊ37Ġ+Ġ
Rollout 88, Act: 9.417
ĠweightsĠwouldĠbeĠ5*5*1*4=100.ĠTheĠbiasesĠareĠadditionalĠparameters
Rollout 7, Act: 9.338
Ġ66NĠ+5NĠ=71N.ĊĊButĠseparately,ĠweĠcanĠsayĠtotalĠpromotedĠscore
Rollout 7, Act: 9.333
59(RĠ-Ġx)Ġ=Ġ71N.ĊĊAdditionally,ĠweĠhaveĠoriginalĠrelations:ĊĊOriginalĠtotal
Rollout 7, Act: 9.209
)Ġ=Ġ76PĠ+Ġ61RĠ=Ġ66NĠ+5NĠ=7
Rollout 32, Act: 9.150
MĠ+ĠM2)Ġ+Ġ669*IĠ=Ġ670*(MĠ+ĠM2
Rollout 41, Act: 9.121
0Ġ+Ġ252Ġ+Ġ576Ġ=927.ĊĊButĠaccordingĠtoĠtheĠall
Rollout 16, Act: 9.112
0*7880Ġ+1953.ĠWait,ĠbutĠlet'sĠverifyĠonceĠmore.ĊĊ
Rollout 58, Act: 9.005
iĠ+Ġ1))^{i}Ġ)Ġ]Ġ)ĠbecauseĠproduct_{jâīłi}Ġ(j/(
Rollout 59, Act: 8.994
^2Ġ+11^2Ġ+4^2=9+100+121
Rollout 67, Act: 8.951
ĠdĠ=Ġ-72*2Ġ+492Ġ=-144Ġ+492=
Rollout 32, Act: 8.909
008}Ġ+ĠM^{2009}ĊĊButĠstartingĠfromĠk=1ĠtoĠk=
Rollout 61, Act: 8.905
ĠofĠ133:Ġ7*19=133.ĠCheckĠ17:
Rollout 59, Act: 8.891
/2)(1/2)(3/7)ĠsinceĠtheĠbiasedĠcoin'sĠprobabilityĠofĠtailsĠisĠ
Rollout 75, Act: 8.890
)^2Ġ+Ġ(2nĠ+1)^2Ġ=Ġ4(k2Ġ+kĠ+m2Ġ+
Top 16 Negative Activations
Rollout 49, Act: -3.458
ĠperfectĠsetĠisĠÏĥ-compact.ĠTherefore,Ġit'sĠaĠunionĠofĠcountablyĠmanyĠcompactĠsets.ĠEach
Rollout 49, Act: -3.472
-compactĠspaceĠwhichĠisĠLindelöfĠisĠaĠcountableĠunionĠofĠcompactĠmetrizableĠspaces...
Rollout 49, Act: -3.525
compact,ĠthenĠinĠparticular,ĠXĠisĠaĠÏĥ-compactĠHausdorffĠspace.ĊĊSoĠXĠ=
Rollout 49, Act: -3.551
ĠisĠsigma-compactĠifĠandĠonlyĠifĠitĠisĠanĠFÏĥĠset.ĠAĠBernsteinĠsetĠisĠnotĠB
Rollout 49, Act: -3.555
ĠcompactĠHausdorffĠspaceĠwithĠnoĠisolatedĠpointsĠisĠuncountable,Ġright?ĠForĠexample,Ġthe
Rollout 49, Act: -3.560
alsĠareĠcompletelyĠmetrizable,ĠhenceĠtheyĠareĠaĠPolishĠspace,ĠhenceĠtheyĠareĠÏĥ-compact?
Rollout 49, Act: -3.598
.ĠSinceĠPĠisĠÏĥ-compact,ĠitĠisĠaĠcountableĠunionĠofĠcompactĠsubsets.ĠEachĠcompactĠsubset
Rollout 49, Act: -3.628
ĠpointĠisĠaĠGδĠset,ĠthenĠmaybeĠXĠisĠcountable.ĊĊWait,ĠinĠaĠHausdor
Rollout 49, Act: -3.654
}Ġ(XĠ-ĠF_{m}),ĠwhichĠisĠaĠGδĠset.ĠIt'sĠasĠweĠhadĠbefore
Rollout 49, Act: -3.676
,ĠinĠcompleteĠmetricĠspaces,ĠtheĠirrationalsĠareĠaĠGδĠset,ĠbutĠcomplementĠofĠGδ
Rollout 49, Act: -3.685
Ġsubspaces,ĠsoĠXĠisĠaĠcontinuousĠimageĠofĠaĠcertainĠspace.ĊĊThisĠseemsĠtooĠvague.ĊĊAlternatively,
Rollout 49, Act: -3.743
ĠeveryĠopenĠsetĠisĠFÏĥ,ĠthenĠtheĠspaceĠisĠhereditarilyĠLindelöf.ĊĊWait,
Rollout 49, Act: -3.890
ĠmetricĠspaces,ĠÏĥ-compactnessĠisĠequivalentĠtoĠbeingĠLindelöfĠandĠsecondĠcountable?ĠWait
Rollout 49, Act: -4.207
ĠallĠpointsĠbeingĠGδ,ĠtheĠĠdiagonalĠisĠaĠĠGδĠsetĠinĠtheĠsquareĠofĠX.
Rollout 49, Act: -4.351
ĠinĠcompleteĠmetricĠspaces,ĠtheĠirrationalsĠareĠaĠGδĠset,ĠbutĠcomplementĠofĠGδĠ(
Rollout 49, Act: -4.397
ĠpointsĠbeingĠGδ,ĠtheĠĠdiagonalĠisĠaĠĠGδĠsetĠinĠtheĠsquareĠofĠX.Ġ

UP_PROJ

Top 16 Positive Activations
Rollout 43, Act: 7.888
-NOTĠwillĠapplyĠanĠXĠgateĠ(i.e.,ĠflipĠtheĠtargetĠqubit)ĠonlyĠifĠtheĠcontrol
Rollout 54, Act: 7.706
ĠtheĠnucleophile.ĠHowever,ĠhydroxideĠisĠaĠstrongerĠbaseĠthanĠpropionateĠ(sinceĠpropionicĠacid
Rollout 42, Act: 7.613
ĠcyclopropaneĠringĠadjacentĠtoĠaĠleavingĠgroup,ĠeliminationĠcanĠleadĠtoĠringĠopening.ĠAlternatively,Ġperhaps
Rollout 54, Act: 7.472
ĠbecauseĠtheĠsolventĠcanĠstabilizeĠchargeĠthroughĠsolvation.ĠForĠexample,ĠinĠpolarĠproticĠsolvents:ĠI
Rollout 43, Act: 7.454
ĠTheĠstandardĠC-NOTĠflipsĠtheĠtargetĠqubitĠ(appliesĠanĠXĠgate)ĠifĠtheĠcontrolĠq
Rollout 54, Act: 7.401
ĠtheĠbetterĠitĠcanĠbreakĠthroughĠsolvationĠandĠattack.ĠAlso,ĠweakerĠbasesĠmightĠbeĠpoorerĠnucleophiles
Rollout 93, Act: 7.372
2*1007)).ĠWait,ĠisĠthat?ĊĊWait,Ġ504ÏĢ/1
Rollout 11, Act: 7.367
,ĠthenĠr3Ġ+1/r3Ġ>2Ġ(ĠasĠAMĠâī¥ĠGM,ĠminimumĠatĠr=1
Rollout 43, Act: 7.361
).ĠTheĠC-NOTĠwillĠapplyĠanĠXĠgateĠ(i.e.,ĠflipĠtheĠtargetĠqubit)Ġonly
Rollout 54, Act: 7.325
ĠlessĠnucleophilicĠbecauseĠtheĠloneĠpairsĠareĠharderĠtoĠaccess,ĠbutĠinĠsolution,ĠmaybeĠalsoĠlessĠsol
Rollout 54, Act: 7.322
ĠcomparingĠdifferentĠanions:ĠbulkierĠonesĠmightĠbeĠlessĠnucleophilicĠbecauseĠofĠstericĠhindrance.
Rollout 71, Act: 7.300
ĠaroundĠ287.2857...ĠwhichĠisĠnotĠanĠinteger.Ġ11?
Rollout 43, Act: 7.254
ĠtheĠstandardĠC-NOT.ĠWait,ĠisĠthisĠaĠC-NOTĠwhereĠtheĠcontrolĠisĠstillĠtheĠfirst
Rollout 43, Act: 7.253
ĠinsteadĠofĠ|1⣩?ĠSoĠit'sĠlikeĠaĠcontrolled-XĠgateĠwithĠcontrolĠstateĠ|0⣩.ĊĊ
Rollout 69, Act: 7.215
Ġtheta?ĠNotĠsure,ĠmaybeĠit'sĠpartĠofĠaĠmultiple-angleĠidentity.ĊĊAlternatively,ĠperhapsĠwritingĠtheĠterm
Rollout 53, Act: 7.208
?ĠMaybeĠquadrilateralĠAEDCĠisĠsplitĠintoĠtwoĠtriangles:ĠAEDĠandĠEDC?ĠWait,
Top 16 Negative Activations
Rollout 97, Act: -2.645
2O5âĨĴ...),Ġthus,ĠtheĠanswerĠderivedĠfromĠconsideringĠtheĠstoiChiorelativeĠderivationĠis
Rollout 6, Act: -2.649
ĠwithĠα_nĠ|ĠnĠâĪĪĠNĠ},ĠthenĠthisĠsetĠisĠsparse.ĊĊToĠretrieveĠα_n,ĠtheĠoracle
Rollout 82, Act: -2.652
021.ĠTotalĠsymbolsâīĪ2.021.ĊĊSoĠworseĠthanĠtheĠall-pairsĠbut
Rollout 24, Act: -2.655
)ĠandĠ(RâĢĻ,ĠCâĢĻ)ĠbeĠminimalĠsaddleĠpairs.ĊĊDefineĠaĠmappingĠf:RâĨĴRâĢĻ
Rollout 46, Act: -2.655
âĪĤU/âĪĤxĊĊd2y/dt2Ġ+Ġ2ĠÏīĠ(dx/dt
Rollout 17, Act: -2.758
ĠmagnitudeĠofĠtheĠintegralĠ$|\mathbf{I}|$Ġ(inĠSIĠunits).Ċ<|im_start|>assistantĊ<|im_start|>
Rollout 82, Act: -2.765
Ġattemps:ĊĊButĠexpectedĠsymbolsâīĪ2.021>2.ĊĊSoĠmaybeĠthereĠareĠcertainĠconstraints
Rollout 65, Act: -2.780
Ġ(1995/c)*(19952/c)=1995^3Ġ/
Rollout 25, Act: -2.789
xĠ[Ġv^4Ġ-Ġ(xĠv^4)/r_p_initial2Ġ+Ġ2Ġv^
Rollout 77, Act: -2.800
ĊTheĠfunctionĠ$f$Ġ,ĠdefinedĠonĠtheĠsetĠofĠorderedĠpairsĠofĠpositiveĠintegers,ĠsatisfiesĠtheĠfollowing
Rollout 25, Act: -2.801
:ĊĊr_p_finalĠ=Ġ[Ġ(r2Ġv2)/μ_finalĠ]Ġ/Ġ[1Ġ+Ġsqrt(
Rollout 65, Act: -2.811
32ÃĹ52ÃĹ72ÃĹ192.ĠTheĠnumberĠofĠdivisorsĠisĠ(2
Rollout 97, Act: -2.813
.ĊĊSinceĠpressureĠisĠproportionalĠtoĠmoles,ĊĊTotalĠpressureĠ=Ġ(n0+Ġ~xi}P0
Rollout 97, Act: -2.814
1-2y).ĊĊButĠrate=Ġd~xi/dt=ĠkĠ(1-2y)ĊĊ
Rollout 99, Act: -2.917
875086175051)=approx2.65731
Rollout 97, Act: -3.240
(initial)=molesĠofĠN2O5}=ĠP0ĠVĠ/ĠRĠT}=ĠAssumingĠclosedĠvessel.ĊĊ

DOWN_PROJ

Top 16 Positive Activations
Rollout 54, Act: 12.741
ĠoppositeĠtoĠbasicityĠbecauseĠtheĠsolventĠcanĠstabilizeĠchargeĠthroughĠsolvation.ĠForĠexample,ĠinĠpolarĠprotic
Rollout 57, Act: 12.208
ĠwhereĠallĠopenĠsetsĠareĠmeasurableĠandĠhaveĠtheĠpropertyĠofĠcontainingĠdensityĠneighborhoodsĠaroundĠeachĠpoint.ĠMoreover,Ġin
Rollout 49, Act: 11.865
ĠsubsetĠisĠFÏĥ,ĠtheĠspaceĠisĠhereditarilyĠLindelöf.ĊĊBecauseĠinĠaĠspaceĠwhere
Rollout 3, Act: 11.846
ĠtermsĠofĠmolarĠmass,ĠsinceĠit'sĠaĠmonatomicĠgas,ĠtheĠmolarĠmassĠwouldĠbeĠapproximately
Rollout 49, Act: 11.541
ĠisĠaĠstrongerĠcondition,ĠperhapsĠcalledĠ"hereditarilyĠFÏĥ"ĠorĠsomething?ĊĊButĠIâĢĻmĠnot
Rollout 57, Act: 11.533
ĠdensityĠpoint,ĠwhichĠisĠpreciselyĠtheĠdefinitionĠofĠbeingĠ"openĠinĠtheĠdensityĠtopology".ĊĊWait,ĠsoĠthis
Rollout 67, Act: 11.462
(3)=9,Ġa=8(3)=24,Ġp=180/3
Rollout 80, Act: 11.454
92.646ĠÃĹĠ10^24ĠgĠ=Ġ9.264
Rollout 49, Act: 11.257
ĠisĠFÏĥ,ĠthenĠtheĠspaceĠisĠhereditarilyĠaĠcountableĠunionĠofĠclosedĠsets,ĠhenceĠin
Rollout 35, Act: 11.146
=-1.ĠThenĠn=18ĠandĠn=23.ĠContradiction.ĊĊCaseĠ2
Rollout 6, Act: 11.104
ĠisĠsimilarĠtoĠproblemĠ1.ĠIfĠyouĠconsiderĠcountablyĠinfiniteĠadviceĠstringsĠforĠallĠn,ĠandĠrepresent
Rollout 15, Act: 11.094
.ĊĊL4:ĠneighborĠL3Ġ(off).ĠRemainsĠoff.ĊĊThus,Ġt=4:ĠO
Rollout 62, Act: 10.864
Ġagain.ĊĊm=24:576Ġ-24Ġ+11=576-
Rollout 24, Act: 10.849
ĠclosureĠoperatorsĠandĠclosedĠsets.ĠMinimalĠpairsĠmayĠcorrespondĠtoĠmeet-irreducibleĠelementsĠinĠtheĠlattice,
Rollout 39, Act: 10.752
789ĠsumsĠto7+8+9=24.ĠThus,ĠtotalĠis342
Rollout 20, Act: 10.668
Ġ*Ġ7.ĊĊ46ĠfactorsĠintoĠ2Ġ*Ġ23.ĊĊSoĠLCM(42
Top 16 Negative Activations
Rollout 58, Act: -10.081
ĠpartĠiĠareĠwrongĠisĠ(i/(i+1))^{iĠ-Ġ1}.ĠForĠtheĠother
Rollout 23, Act: -10.091
6/(5.88)=0.1178837216ĠsâĪĴ
Rollout 69, Act: -10.142
_i)Ġ=Ġ[1Ġ-Ġcos(2Ġtheta_i)]Ġ/Ġ[2Ġcos(3Ġtheta_i)]ĊĊ
Rollout 60, Act: -10.253
Ġ6,ĠsinceĠ82*6Ġ=Ġ492,ĠsoĠ492Ġ/
Rollout 87, Act: -10.262
(r,θ)Ġ[Ġ(1/r)Ġ*Ġr2ĠsinθĠdrĠdθĠdÏĨĠ]ĊĊBut
Rollout 99, Act: -10.361
75Ġ-Ġ(-0.39794))/(1.1575Ġ+0.
Rollout 17, Act: -10.433
ĠweĠintegrateĠoverĠallĠanglesĠ(sinceĠdVĠ=Ġr2ĠsinθĠdθĠdÏĨĠdr),Ġwe
Rollout 17, Act: -10.627
ĠintegratingĠtheĠmagneticĠfieldĠoverĠtheĠprobabilityĠdensityĠ|Ψ(r)|2.ĠSinceĠtheĠelectron'sĠwavefunctionĠis
Rollout 23, Act: -10.646
1/5.88âīĪ0.1179ĠsâĪĴ1ĊĊThenĠlambda_total=
Rollout 38, Act: -10.672
1Ġ+7/24)=Ġsqrt(1Ġ+49/576)/Ġ(31
Rollout 23, Act: -10.857
ĠtheĠpreviousĠstepsĠthatĠIĠcanâĢĻtĠsee,ĠbutĠeachĠstepĠseemsĠcorrect.ĠSoĠlikely,ĠanswerĠisĠB
Rollout 17, Act: -10.873
ĠbyĠtheĠprotonĠandĠtheĠprobabilityĠdensityĠ|Ψ(r)|2ĠofĠtheĠelectronĠinĠtheĠgroundĠstateĠofĠa
Rollout 17, Act: -10.957
_p]ĠdividedĠbyĠr3ĠmultipliedĠbyĠ|Ψ(r)|2,ĠthenĠintegratedĠoverĠallĠspace.ĊĊBut
Rollout 69, Act: -11.338
^\inftyĠ[1Ġ-Ġcos(2t_i)]/(2Ġcos(3t_i))ĊĊNote
Rollout 69, Act: -11.548
ĠourĠtermĠbecomesĠ[1Ġ-Ġcos(2t_i)]/(2Ġcos(3t_i))ĊĊLet
Rollout 81, Act: -11.723
ĠBAC)=Ġ[tan(angleĠBAD)+tan(angleĠCAD)]/[1Ġ-tan(angleĠBAD)tan(angle

Layer 60

GATE_PROJ

Top 16 Positive Activations
Rollout 76, Act: 1.901
ĠthatĠdoesĠnotĠhaveĠaĠ2-by-2ĠredĠsquareĠisĠ$\fracĠ{m}{n}$Ġ,
Rollout 42, Act: 1.877
,4,5,6-hexahydropentaleneĊ(D)Ġspiro[3.
Rollout 57, Act: 1.669
ĠaĠdensityĠpoint,ĠwhichĠisĠpreciselyĠtheĠdefinitionĠofĠbeingĠ"openĠinĠtheĠdensityĠtopology".ĊĊWait,Ġso
Rollout 5, Act: 1.600
zoidĠintoĠtwoĠregionsĠwhoseĠareasĠareĠinĠtheĠratioĠ$2:Ġ3$Ġ.ĠLetĠ$x
Rollout 42, Act: 1.538
4,5,6-hexahydropentaleneĊ(D)Ġspiro[3.4
Rollout 65, Act: 1.504
ĠandĠbothĠofĠwhichĠhaveĠnonzeroĠvolume.ĠĠGivenĠthatĠ$b=1995,$ĠforĠhow
Rollout 74, Act: 1.484
,ĠandĠK_sĠisĠtheĠcoefficientĠforĠtheĠheelingĠerrorĠcorrection.ĠButĠwait,ĠmaybeĠK_bĠand
Rollout 54, Act: 1.395
-olate,Ġ2.ĠHydroxide,Ġ3.ĠPropionate,Ġ4.ĠMethanol
Rollout 49, Act: 1.277
ĠitĠisĠfirstĠcountable?ĠOrĠmaybeĠhasĠcountableĠextent?ĠButĠI'mĠnotĠsure.ĊĊAlternatively,
Rollout 15, Act: 1.270
".ĠTherefore,ĠtheĠlamp'sĠownĠstateĠatĠtĠdoesn'tĠmatter.ĠTheĠnextĠstateĠdependsĠonlyĠonĠthe
Rollout 32, Act: 1.262
Ġ\end{pmatrix}.\]FindĠtheĠsumĠofĠtheĠentriesĠofĠ$\mathbf{M}
Rollout 49, Act: 1.240
öfĠspace,ĠeachĠclosedĠsetĠhasĠaĠcountableĠsubsetĠdenseĠinĠitĠ(sinceĠscatteredĠimpliesĠright-separated
Rollout 49, Act: 1.193
Alternatively,ĠuseĠtheĠfactĠthatĠXĠhasĠaĠcountableĠextentĠ(everyĠclosedĠdiscreteĠsubspaceĠisĠcountable
Rollout 49, Act: 1.136
-countable,ĠletĠmeĠassumeĠitĠhasĠaĠcountableĠbase.ĊĊWait,Ġno.ĠForĠexample,Ġthe
Rollout 77, Act: 1.105
ĠonĠtheĠsetĠofĠorderedĠpairsĠofĠpositiveĠintegers,ĠsatisfiesĠtheĠfollowingĠproperties:Ġ\[f(x,Ġx
Rollout 65, Act: 1.102
ĠcutsĠ$P_{}$ĠintoĠtwoĠprisms,ĠoneĠofĠwhichĠisĠsimilarĠtoĠ$P_{},
Top 16 Negative Activations
Rollout 69, Act: -10.435
Ġ/12âĪļ3Ġ)Ġ=Ġ(6Ġ-3âĪļ3Ġ)/(12âĪļ3Ġ)ĊĊBut
Rollout 95, Act: -10.473
ĠisĠatĠB,ĠbutĠ92Ġ+Ġ(5âĪļ3)2Ġ=Ġ81Ġ+Ġ
Rollout 12, Act: -10.483
Ġt2)ĊĊSoĠdenominatorĠbecomesĠsqrt(Ġ(Ġ(t2Ġ-Ġν2)/ν2Ġ)Ġ(1
Rollout 11, Act: -10.504
1.ĠThenĠabc=ĠÏīĠkĠ*ĠÏī2ĠkĠ*1/(k2ĠÏī3
Rollout 69, Act: -10.506
/(12âĪļ3Ġ)ĊĊButĠ6Ġ-3âĪļ3Ġ)/(12âĪļ3Ġ)=
Rollout 36, Act: -10.531
^2Ġ+Ġz^2]Ġ=Ġ60âĪļ10Ċ2.ĠDistanceĠfromĠPĠto
Rollout 69, Act: -10.557
3Ġ/3Ġ-1Ġ)/4=Ġ(2âĪļ3Ġ-3Ġ)/(12Ġ).ĊĊTherefore
Rollout 17, Act: -10.565
âĪ«_{-1}^{1}Ġ(3u2Ġ-1)Ġ(-du)Ġ=ĠâĪ«
Rollout 65, Act: -10.584
1995^3/(1995^4/a2)=Ġa2Ġ/19
Rollout 1, Act: -10.634
Ġ=Ġ2dĠ/Ġsqrt(2)Ġ=ĠdĠsqrt(2)Ġ>Ġd,ĠsoĠtheĠspheres
Rollout 69, Act: -10.643
âĪļ3Ġ)/(4*3Ġ)=Ġ(2âĪļ3Ġ-3Ġ)Ġ/Ġ12.
Rollout 95, Act: -10.673
âĪļ3)/2Ġ=Ġ(21Ġ+Ġ5âĪļ3)/2.ĠHmm,ĠseemsĠcomplicated.
Rollout 14, Act: -10.762
):ĊĊλ2Ġλ=1/12âĨĴλ3=1/12âĨĴλ=(1
Rollout 95, Act: -11.061
ĠisĠ12Ġunits,ĠandĠBCĠisĠ5âĪļ3.ĠHmm,ĠtheĠsidesĠareĠinteresting.
Rollout 32, Act: -11.233
2)Ġ+Ġ(-Ïī)(1)Ġ=ĠÏī4Ġ+Ġ(-Ïī).ĠButĠÏī3Ġ=
Rollout 17, Act: -11.463
1}^1ĠduĠ=Ġ3Ġ[Ġ(u3/3)ĠfromĠ-1ĠtoĠ1

UP_PROJ

Top 16 Positive Activations
Rollout 14, Act: 9.070
2Ġdx)Ġ*Ġsqrt(âĪ«12Ġdx)Ġ)Ġ=Ġsqrt(J)Ġ*Ġsqrt(âĪ«1
Rollout 14, Act: 8.888
}Ġ(âĪ«ĠxĠdxĠ)^{3/2}Ġ?ĠButĠagain,ĠâĪ«xdxĠdiverges
Rollout 81, Act: 8.728
Ġ/Ġ(b2Ġ+Ġc2Ġ-a2)ĠwhereĠÎĶĠisĠarea,ĠbutĠperhapsĠmoreĠstraightforwardĠis
Rollout 37, Act: 8.712
,ĠwhichĠisĠdivisibleĠbyĠ2015!ĠwhenĠevaluatedĠatĠintegerĠarguments?ĠWait,ĠbutĠlet
Rollout 24, Act: 8.695
ĠinĠtermsĠofĠtheĠrelationĠdefinedĠbyĠdomination.ĠRĠisĠaĠminimalĠsetĠsuchĠthatĠeveryĠrowĠisĠeitherĠin
Rollout 24, Act: 8.684
ĠR.ĊĊMinimalityĠimpliesĠthatĠRĠandĠCĠareĠminimalĠsuchĠdominatingĠsets.ĊĊButĠinĠbipartite
Rollout 24, Act: 8.663
ĠensuringĠtheyĠallĠhaveĠtheĠsameĠsize.ĠForĠinstance,ĠinĠaĠbipartiteĠgraphĠwithĠaĠperfectĠmatching
Rollout 37, Act: 8.600
Ġa_j),ĠwhichĠinĠturnĠisĠdivisibleĠbyĠn!Ġ?ĊĊWait,ĠifĠweĠhaveĠthatĠdet(A)
Rollout 24, Act: 8.574
ĠsameĠsizes.ĠMaybeĠuseĠDilworth'sĠtheoremĠorĠsimilar.ĊĊAlternatively,ĠthinkĠinĠtermsĠofĠtheĠtwoĠminim
Rollout 14, Act: 8.567
1^2Ġdx)(âĪ«f^2Ġdx)ĠbyĠCauchy-Schwarz,ĠbutĠas
Rollout 28, Act: 8.543
ĠtrivialĠwhereĠbothĠareĠzero,ĠbutĠthatĠgivesĠaĠline,ĠnotĠaĠconic).ĠLet'sĠcheck.ĊĊ
Rollout 14, Act: 8.509
Ġ=Ġsqrt(J)Ġ*Ġsqrt(âĪ«1Ġdx)Ġ=Ġsqrt(1)Ġ*âĪŀĠ=âĪŀ.
Rollout 24, Act: 8.465
ĠR.ĊĊTherefore,ĠmaybeĠtheĠsizeĠofĠRĠisĠtheĠminimalĠnumberĠofĠrowsĠneededĠtoĠcoverĠallĠrowsĠunder
Rollout 24, Act: 8.439
ĠC)ĠtoĠbeĠaĠsaddleĠpair,ĠRĠmustĠbeĠaĠdominatingĠsetĠforĠallĠrowsĠinĠtheĠrowĠdominance
Rollout 37, Act: 8.414
ĠalgebraicĠconstructionĠinherentlyĠhasĠ2015!ĠasĠaåĽłåŃIJãĢĤĠButĠInĠtheĠalgebraicĠscenario
Rollout 5, Act: 8.350
ĠInĠaĠtrapezoid,ĠtheĠmidlineĠ(theĠsegmentĠconnectingĠtheĠmidpointsĠofĠtheĠlegs)
Top 16 Negative Activations
Rollout 36, Act: -2.348
2ĊĊFirst,ĠcomputeĠdiscriminant:ĊĊÎĶ=30,6242Ġ-4*1
Rollout 66, Act: -2.363
,Ġsomewhere,ĠmyĠreasoningĠmustĠbeĠwrong.ĊĊWait,ĠbutĠmĠandĠnĠmustĠbeĠofĠoppositeĠparity.
Rollout 25, Act: -2.364
ĠEarth,ĠsoĠthereĠmustĠbeĠanĠanswer.ĠHence,ĠthereĠmustĠbeĠanĠerrorĠinĠreasoning.ĊĊWait.
Rollout 30, Act: -2.392
1}=\left(a_{n-1}-1\right)\left(b_{n}+1
Rollout 7, Act: -2.402
.ĠSoĠIĠmustĠhaveĠmadeĠaĠmistake.ĊĊWait,ĠalternativeĠapproach.ĠLetĠmeĠuseĠanotherĠangle.ĊĊGiven
Rollout 7, Act: -2.411
Ġlikely,ĠtheĠproblemĠexpectsĠthereĠtoĠbeĠsolutions.ĠSoĠIĠmustĠhaveĠmadeĠaĠmistake.ĊĊWait,Ġalternative
Rollout 95, Act: -2.474
ĠaskĠforĠsuchĠanĠanswerĠifĠitĠwereĠzero.ĊĊSo,Ġclearly,Ġthere'sĠanĠerrorĠinĠthisĠreasoning
Rollout 90, Act: -2.515
14Ġmod2018âĩĴL=2018-1,014=
Rollout 23, Act: -2.532
ĠisĠ~8.ĠSomething'sĠwrongĠhere.ĊĊWait,Ġwait.ĠAnĠalternativeĠapproach:ĠmaybeĠtheĠtermĠ"
Rollout 39, Act: -2.620
k=1}^{321}Ġ10^kĠ=Ġ10Ġ\frac{
Rollout 89, Act: -2.687
.ĠTheĠproblemĠwouldn'tĠgiveĠimpossibleĠnumbers.ĠTherefore,ĠperhapsĠIĠmadeĠaĠmiscalculation.ĊĊWait
Rollout 90, Act: -2.718
16Ġmod2018âĩĴĠL=2018-1,016=
Rollout 90, Act: -2.840
.ĊĊ2018*45=Ġ2018*40=80,
Rollout 95, Act: -3.099
ĠforĠsuchĠanĠanswerĠifĠitĠwereĠzero.ĊĊSo,Ġclearly,Ġthere'sĠanĠerrorĠinĠthisĠreasoning.ĊĊ
Rollout 90, Act: -3.212
Ġmod2018.ĠTherefore,ĠL=2018Ġ-1,012=
Rollout 71, Act: -3.281
02=400.ĠthenĠleftĠ4022Ġ-400=36

DOWN_PROJ

Top 16 Positive Activations
Rollout 77, Act: 13.617
*13=Ġ4*91=Ġ364.ĠYes.ĠSoĠyes,Ġf
Rollout 96, Act: 13.181
·28,561âīĪ3.11ĊĊButĠpreciseĠcomputation:ĊĊ28,5
Rollout 38, Act: 13.021
403/217âīĪ1.857x.ĊĊComparingĠwithĠW=(2
Rollout 28, Act: 13.020
ĠâĪļ2/3Ġ)2Ġ/Ġ(1/18)Ġ+ĠY2/(1/6
Rollout 70, Act: 13.005
0Ġ/7^3=2400/343âīĪ6.997,
Rollout 96, Act: 12.974
=ĠDivideĠnumeratorĠandĠdenominatorĠbyĠ12:Ġ193/169.ĠSimilarly,
Rollout 37, Act: 12.623
125Ġ-Ġ27*25Ġ=Ġ1125Ġ-Ġ675
Rollout 62, Act: 12.298
=1100Ġ-10+1=1091.ĠPrime.ĊĊHmm.ĠSo
Rollout 3, Act: 12.291
Ġ0.697ĠâīĪ28,624;ĠsqrtĠâīĪ169
Rollout 65, Act: 12.280
Ġ/3=3*25*49*361=3*52*7
Rollout 22, Act: 12.238
60539Ġ~Ġ0.0255.ĠSoĠtotalĠapproxĠ3.3
Rollout 28, Act: 12.235
36k2ĊĊ-108hĠ-108kĊĊ+81=0
Rollout 70, Act: 12.210
Ġ(>333),ĠsoĠ7^3=343Ġ(>333,ĠsoĠ
Rollout 27, Act: 12.118
94e-26ĠâīĪ2.228e10Ġm2/s2.
Rollout 70, Act: 11.988
i=3:1000Ġ-2*343=1000Ġ-6
Rollout 70, Act: 11.986
ĠinĠbase-7:ĠSinceĠ7^3=343,Ġ363-3
Top 16 Negative Activations
Rollout 87, Act: -13.667
ĊĊCalculate:ĊĊ3*log(1Ġ+Ġsqrt(2))ĠâīĪĠ3Ġ*Ġ0.8
Rollout 21, Act: -13.894
=-1,ĠwhichĠdoesĠnotĠsatisfyĠtheĠequation.ĊĊHenceĠsuchĠaĠmodifiedĠfunctionĠwouldnâĢĻtĠworkĠasĠz
Rollout 22, Act: -14.068
Ġ1,928ĠâīĪĠ1,777Ġm/s.ĠCloseĠtoĠ1
Rollout 22, Act: -14.331
Ġ*Ġ0.921Ġ=ĠapproxĠ1,770Ġm/s,ĠwhichĠwouldĠcorrespond
Rollout 3, Act: -14.356
016÷0.697âīĪ27,950;Ġsqrt(27
Rollout 81, Act: -14.565
Ġ[20/h]/[1Ġ-51/h2]Ġ=Ġ[20/h]/[(h
Rollout 3, Act: -14.933
/0.697433âīĪ27,950.Ġsqrt(27
Rollout 3, Act: -14.943
/0.697433âīĪ27,950ĊĊsqrt(27
Rollout 1, Act: -15.314
ĠballsĠareĠnotĠcompactĠhenceĠthereĠisĠnoĠguarantee.ĊĊHencesĠperhapsĠrequiresĠanotherĠapproach,ĠusingĠtheĠstructureĠof
Rollout 68, Act: -15.631
)=sqrt(254000).ĠHmmm.ĊĊButĠregardless,ĠtheĠapproximateĠcalculation:ĊĊsqrt
Rollout 14, Act: -15.717
Ġright-handĠsideĠisĠ8ĠtimesĠtheĠintegralĠofĠfĠsquaredĠtimesĠintegralĠofĠxĠf.ĠGivenĠthatĠintegr
Rollout 14, Act: -15.747
ĠsomeĠsortĠofĠinnerĠproductĠinvolvingĠxĠf(x).ĠHmmm.ĊĊWait,ĠwhatĠifĠweĠmakeĠtheĠsubstitution
Rollout 17, Act: -16.483
ĠbyĠtheĠprotonĠandĠtheĠprobabilityĠdensityĠ|Ψ(r)|2ĠofĠtheĠelectronĠinĠtheĠgroundĠstateĠofĠa
Rollout 65, Act: -16.562
ĠrestĠcausesĠaĠsquareĠlaw.ĊĊButĠmyĠmemoryĠisĠharkeningĠtoĠaĠmathĠconceptĠwhereĠbreakingĠaĠrectangle
Rollout 22, Act: -16.671
68.7768Ġ*Ġ157.93.ĠSoĠ18,
Rollout 23, Act: -17.376
1/5.88âīĪ0.1179ĠsâĪĴ1ĊĊThenĠlambda_total=

Layer 61

GATE_PROJ

Top 16 Positive Activations
Rollout 66, Act: 10.525
4ĠandĠ13:Ġgcd(84,13)=13ĠfactorsĠ13;
Rollout 66, Act: 10.444
2,ĠweĠgetĠtheĠprimitiveĠtripletĠ(84,13,85Ġ).ĠThus,ĠcheckĠif
Rollout 66, Act: 10.326
ĠbeĠdividedĠbyĠ2ĠtoĠproduceĠ(84,13,85Ġ).ĠButĠthenĠhowĠdoes
Rollout 65, Act: 10.228
,ĠpĠdivides1995*q.ĠSinceĠpĠandĠqĠareĠcoprime,ĠthenĠpĠdivides
Rollout 19, Act: 10.138
17ĠmodkĠâĩĴĠ83âī¡0ĠmodkâĩĴĠthusĠkĠdivides83.ĊĊAs8
Rollout 66, Act: 10.138
ĠdownĠbyĠ2,ĠweĠhaveĠ(84,13,85Ġ).ĠLet'sĠcheckĠ"
Rollout 0, Act: 10.107
ĠtotalĠpairs,ĠsoĠ256/2Ġ=128.ĠHenceĠ128Ġsuch
Rollout 19, Act: 10.107
Ġk.ĠSoĠneedĠx2Ġâī¡17Ġmodk.ĠSoĠneedĠcongruenceĠtoĠholdĠfor
Rollout 62, Act: 10.106
.ĠButĠthisĠoneĠisĠm^2Ġ-mĠ+11.ĠFromĠm=1ĠtoĠ1
Rollout 62, Act: 10.096
ĠpositiveĠintegerĠmĠsuchĠthatĠm^2Ġ-mĠ+11ĠisĠaĠproductĠofĠatĠleastĠfourĠnot
Rollout 19, Act: 10.080
2.ĊĊWeĠneedĠx2Ġâī¡17ĠmodĠk.ĊĊButĠalso,ĠnoteĠthatĠcongruence
Rollout 66, Act: 10.072
ĠtheĠrequiredĠequation.ĠHence,Ġperimeter=84Ġ+13Ġ+85=182.
Rollout 65, Act: 10.047
Ġdivides1995*q.ĠSinceĠpĠandĠqĠareĠcoprime,ĠthenĠpĠdivides19
Rollout 19, Act: 9.996
4)/2]ĠmodkĠâī¡10ĠmodkĊĊTherefore,ĠtheĠexpressionĠisĠequivalentĠtoĠ
Rollout 65, Act: 9.994
ĠC2/BĠ<=B,ĊĊâĩĴC2/BĠ<=B,ĊĊâĩĴĠC2Ġ<=B2,ĊĊâĩĴ
Rollout 19, Act: 9.981
7ĠmodkâĩĴ100âī¡17ĠmodkâĩĴĠ83âī¡0modkâĩĴ
Top 16 Negative Activations
Rollout 66, Act: -2.315
[ĊĠĠĠ\frac{ab}{cĠ-Ġa}Ġ=Ġ6pĊĠĠĠ\]ĊĠĠĠwhere
Rollout 5, Act: -2.320
Ġ+Ġ25}{bĠ+Ġ75}Ġ=Ġ\frac{2}{3}Ċ\]Ċ
Rollout 27, Act: -2.428
2}ĠmĠv_{z,Ġ\text{rms}}^2ĊĠĠĠĠĠ\]ĊĠĠĠ-ĠSol
Rollout 68, Act: -2.431
ĠpropertiesĠofĠlogarithms:Ċ\[Ċ\log(kx)Ġ=Ġ2\log(x+2)
Rollout 89, Act: -2.440
Ġ\[ĊĠĠĠ\frac{3}{5}TĠ=Ġ100ĊĠĠĠ\]Ċ4
Rollout 59, Act: -2.455
28}Ġ+Ġ\frac{4}{28}Ġ=Ġ\frac{10}{28
Rollout 55, Act: -2.477
}{1}.\]ComputeĠ$\mathbf{A}^{2018}.$Ċ<|im_start|>
Rollout 55, Act: -2.491
Ġ\(12Ġ\timesĠ\frac{\pi}{6}Ġ=Ġ2\pi\)),ĠandĠthe
Rollout 72, Act: -2.505
ĠsumĠisĠ\(Ġ1Ġ+Ġ1Ġ+Ġ(-1)Ġ=Ġ1Ġ\).ĊĠĠĠ-ĠThisĠconstruction
Rollout 53, Act: -2.589
Ġ=Ġ2.5\),ĠweĠfindĠ\(ACĠ=Ġ5\).ĊĊ3.Ġ**SettingĠup
Rollout 5, Act: -2.669
bĠ+Ġ25}{bĠ+Ġ75}Ġ=Ġ\frac{2}{3}Ċ\
Rollout 66, Act: -2.680
Ġ(84Ġ+Ġ13Ġ+Ġ85)Ġ=Ġ6Ġ\timesĠ182
Rollout 35, Act: -2.706
ĠnĠ+Ġ3Ġ\)ĠtranslatesĠtoĠ\(ĠQ(n)Ġ=Ġ0Ġ\),ĠmeaningĠ\(Ġn_
Rollout 52, Act: -2.733
Ġ+Ġ\frac{k(k-1)}{2}Ġ=Ġ2014Ċ\]ĊR
Rollout 55, Act: -2.841
12Ġ\timesĠ\frac{\pi}{6}Ġ=Ġ2\pi\)),ĠandĠtheĠreflection
Rollout 52, Act: -2.913
ĠnĠ+Ġ\frac{k(k-1)}{2}Ġ=Ġ2014Ċ\]Ċ

UP_PROJ

Top 16 Positive Activations
Rollout 29, Act: 4.803
Ġintuitively,ĠtheĠpressureĠthereĠmightĠjustĠbeĠrho*g*h.ĠButĠsinceĠtheĠhandleĠhasĠaĠhigherĠmilk
Rollout 29, Act: 4.788
ĠpressureĠatĠtheĠbottomĠofĠtheĠhandleĠwouldĠbeĠrho*g*H,ĠbutĠifĠtheĠmainĠjugĠisĠconnected
Rollout 29, Act: 4.682
ĠisĠrho*g*hĠandĠtheĠhandle'sĠisĠrho*g*H.ĠButĠtheĠproblemĠisĠaskingĠaboutĠthe
Rollout 29, Act: 4.649
'sĠbottomĠ(ifĠconsideredĠseparately)ĠwouldĠhaveĠrho*g*H.ĠButĠtheĠquestionĠisĠaboutĠtheĠgauge
Rollout 29, Act: 4.590
ĠofĠtheĠmainĠjugĠwouldĠbeĠatmosphericĠpressureĠ+Ġrho*g*h.ĠSinceĠit'sĠopen,ĠgaugeĠpressureĠis
Rollout 29, Act: 4.543
ĠpressureĠatĠtheĠbottomĠofĠtheĠmainĠjugĠisĠrho*g*h.ĠHowever,ĠmaybeĠbecauseĠtheĠhandle'sĠmilk
Rollout 29, Act: 4.514
ĠmainĠjug'sĠbottomĠhasĠaĠgaugeĠpressureĠofĠrho*g*h,ĠwhileĠtheĠhandle'sĠbottomĠ(ifĠconsidered
Rollout 29, Act: 4.506
ĠpressureĠofĠtheĠsealedĠairĠinĠtheĠhandleĠplusĠrho*g*H.ĠSinceĠtheĠpressureĠatĠtheĠinterfaceĠbetween
Rollout 29, Act: 4.503
*H.ĠThenĠtheĠgaugeĠpressureĠwouldĠbeĠrho*g*HĠ-ĠatmosphericĠpressure.ĠButĠsinceĠtheĠmain
Rollout 25, Act: 4.475
.ĠSimilarly,ĠaccelerationĠonĠEarthĠisĠF/MĠ=ĠG*mĠ/Ġd^2,ĠwhichĠwouldĠcauseĠEarth
Rollout 29, Act: 4.464
Ġequal:ĊĊrho*g*hĠ+ĠatmosphericĠpressureĠ=Ġrho*g*HĠ+ĠP_handleĊĊTherefore,ĠP_handle
Rollout 29, Act: 4.440
ĠpressureĠatĠtheĠbottomĠofĠtheĠhandleĠwouldĠbeĠrho*g*H.ĠThenĠtheĠgaugeĠpressureĠwouldĠbeĠrho
Rollout 29, Act: 4.426
Ġatmospheric,ĠwouldĠbeĠ(atmosphericĠ+Ġrho*g*h)Ġ-ĠatmosphericĠ=Ġrho*g*hĠinĠthe
Rollout 29, Act: 4.384
Ġjug,ĠwhichĠisĠh,ĠsoĠPĠ=Ġrho*g*h.ĠHowever,ĠifĠtheĠhandleĠisĠsomehowĠstill
Rollout 48, Act: 4.380
Ġvalid,ĠindicatingĠnoĠerrorĠinĠtheĠheader.ĊĊ\boxed{No}<|im_end|>
Rollout 29, Act: 4.343
ForĠtheĠmainĠjug:ĠP_gaugeĠ=Ġrho*g*hĊĊForĠtheĠhandle:ĠP_gaugeĠ=
Top 16 Negative Activations
Rollout 17, Act: -7.581
/s2,ĠbutĠhereĠinĠanyĠcase,ĠtheĠunitsĠneedĠtoĠresolve,ĠbutĠsinceĠtheĠpreviousĠcalculationĠshowed
Rollout 30, Act: -7.597
?ĠWait,ĠmaybeĠnot.ĠItĠdependsĠifĠthereĠisĠaĠmiscalculationĠhere.ĊĊLetĠmeĠcheck
Rollout 73, Act: -7.612
Ġquickly,ĠbutĠmoduloĠ1000ĠcouldĠhelp.ĊĊWait,ĠmaybeĠverifyingĠN=163
Rollout 33, Act: -7.621
Ġ-48Ġ(m=3).ĠLetĠmeĠcheck:ĊĊ720Ġ-Ġ372Ġ=
Rollout 73, Act: -7.630
16398.ĠAlternatively,ĠperhapsĠIĠmadeĠaĠmistakeĠinĠcalculation.ĊĊEarlierĠsteps:ĊĊWeĠhad
Rollout 16, Act: -7.641
4818...ĊĊWait,ĠbutĠperhapsĠthat'sĠgettingĠtooĠapproximate.ĠLet'sĠcomputeĠeachĠfractionĠexactly
Rollout 30, Act: -7.677
ĠWait,ĠmaybeĠnot.ĠItĠdependsĠifĠthereĠisĠaĠmiscalculationĠhere.ĊĊLetĠmeĠcheckĠthrough
Rollout 93, Act: -7.716
/N)]^{2N}.ĠWait,ĠisĠthereĠaĠknownĠformulaĠforĠthisĠkindĠofĠsum?ĊĊAlternatively,
Rollout 2, Act: -7.723
izingĠeachĠterm,ĠmightĠuseĠmodulus1000.ĠHowever,ĠsinceĠ1000=
Rollout 48, Act: -7.755
.ĠSoĠF946).ĊĊButĠwait,ĠmaybeĠhandlingĠitĠinĠhex:ĊĊ0x450
Rollout 76, Act: -7.771
ĠButĠthatĠmightĠbeĠcomplicated.ĠAlternatively,ĠperhapsĠthere'sĠaĠmistakeĠinĠinclusion-exclusion.ĊĊAlternatively,Ġthink
Rollout 73, Act: -7.806
0=398.ĊĊButĠmightĠweĠhaveĠmadeĠaĠmiscalculationĠwhenĠcomputingĠsum_{i
Rollout 48, Act: -7.819
x10000.ĠButĠIĠcanĠdoĠtheĠsum:ĊĊ4500ĊĊ+B6
Rollout 89, Act: -7.874
ĠmustĠbeĠaĠwholeĠnumber,ĠsoĠseemsĠlikeĠthereĠisĠinconsistencyĠhere.ĊĊWait,ĠbutĠmaybeĠtheĠproblemĠis
Rollout 72, Act: -7.881
.ĠGoingĠthroughĠprimes,ĠbutĠmaybeĠweĠcanĠfindĠaĠpattern.ĊĊSoĠsoĠfar,Ġn=2,
Rollout 33, Act: -7.887
)Ġ-48Ġ(m=3).ĠLetĠmeĠcheck:ĊĊ720Ġ-Ġ372

DOWN_PROJ

Top 16 Positive Activations
Rollout 65, Act: 13.694
1995,Ġk*c].ĊĊPossiblyĠre-ordered:ĊĊSupposeĠoriginalĠorderedĠ[a,
Rollout 48, Act: 13.090
=3,609ĠremĠ14ĠâĨĴĠE.Ġ3,609Ġ/1
Rollout 91, Act: 13.026
ĠtheĠvalueĠofĠ\(|\cotĠBĠ-Ġ\cotĠC|\)ĠisĠ\(\boxed{2}\).
Rollout 25, Act: 12.721
Ġ1.008e13ĊĊNumerator:Ġ1.008e1
Rollout 6, Act: 12.645
ĠlogĠnĠ+Ġn^kĠâīĪĠm?ĠUnlikely.ĠForĠexample,ĠmĠisĠdominatedĠbyĠn
Rollout 81, Act: 12.460
1/h2Ġ)ĊĊĠWithh=11:ĊĊNume:20/11.ĊĊĠDenom
Rollout 81, Act: 12.315
,Ġno.ĊĊWait,ĠsinA=2ÎĶ/(bc).ĊĊcosA=(b2Ġ+c2
Rollout 81, Act: 12.304
ĠsinA.ĊĊĠthusĠsinAĠ=Ġ2ÎĶ/(bc).ĊĊLawĠofĠCosines:Ġa2=
Rollout 23, Act: 12.262
ButĠln(1.9e31)=nearlyĠ72.ĠTherefore,Ġexp(7
Rollout 37, Act: 12.227
!ĠasĠaĠfactor.ĠButĠIĠcannotĠexplainĠitĠconcretely.ĊĊAlternatively,ĠconsiderĠthatĠtheĠdeterminantĠcan
Rollout 84, Act: 12.116
MultiplyĠthroughĠbyĠ18:Ġ9Ġ-Ġ18xĠ=Ġ4*(1Ġ-Ġx
Rollout 22, Act: 11.880
.002016Ġkg/molĊĊNumerator:Ġ8Ġ*ĠRĠ*ĠTĠ=8
Rollout 16, Act: 11.876
ĠwhichĠmightĠbeĠcomplicated.ĠHowever,ĠmaybeĠweĠcanĠreindexĠit.ĠLetĠmĠ=ĠkĠ-1.
Rollout 11, Act: 11.871
Ġ\overline{ab}Ġ+Ġ\overline{bc}Ġ+Ġ\overline{ca}Ġ=
Rollout 81, Act: 11.745
121/11âīĪ11ãĢĤĊĊĠConfusionĠagain!ĊĊĠThusĠTwoĠdifferentĠequationsĠarise:ĊĊFrom
Rollout 25, Act: 11.369
.49e6Ġm2/s2ĊĊNumerator:Ġ10.08e1
Top 16 Negative Activations
Rollout 33, Act: -17.215
kmĠ-Ġkm-1)!Ġ*Ġ(6Ġ-Ġkm)!.ĊĊHence,ĠforĠinclusion-exclusion,
Rollout 36, Act: -17.253
2Ġ=Ġ18000Ġ--Ġ(2')ĊĊEquationĠ(3):ĠSĠ-Ġ2
Rollout 36, Act: -17.315
ĠusĠsubtractĠequationsĠ(1'),Ġ(2'),Ġ(3')ĠtoĠfindĠrelationships.ĊĊFromĠ(1')Ġ-
Rollout 0, Act: -17.404
Ġ20!ĠorĠaĠ<Ġsqrt(20!).ĠTherefore,ĠtheĠnumberĠofĠsuchĠaĠisĠhalf
Rollout 36, Act: -17.659
4')ĊĊLetĠusĠsubtractĠequationsĠ(1'),Ġ(2'),Ġ(3')ĠtoĠfindĠrelationships.ĊĊFromĠ(
Rollout 33, Act: -18.348
k4Ġ-k3)!*(6Ġ-k4)!.ĊĊInĠotherĠwords,ĠeachĠconsecutiveĠdifference.
Rollout 36, Act: -18.703
ĠasĠbefore)ĊĊSimilarly,Ġ(1')Ġ-Ġ(3'):Ġ-2sĠxĠ+Ġ2sĠz
Rollout 36, Act: -18.811
Ġ=Ġ-864Ġ=>ĠsubstituteĠintoĠEquationĠ(1'):ĊĊSĠ+Ġ(-864)Ġ=
Rollout 24, Act: -19.049
,Ġa(r,Ġc)Ġâī¤Ġa(r,Ġc').ĊĊSimilarly,ĠFromĠtheĠsaddleĠpairĠ(R
Rollout 36, Act: -19.204
Ġ-Ġs2ĊĊSoĠsubstitutingĠintoĠequationĠ(4'):ĊĊ36000Ġ+Ġ2
Rollout 36, Act: -19.694
,ĠifĠweĠsubtractĠequationsĠ(2')Ġ-Ġ(3'):Ġ-2sĠyĠ+Ġ2sĠz
Rollout 36, Act: -20.356
ĠfromĠanĠequationĠ(1').ĊĊFromĠequationĠ(1'):ĊĊSĠ=Ġ36000
Rollout 36, Act: -21.008
'):ĊĊEquationĠ(d)Ġ+ĠequationĠ(1'):ĊĊ(SĠ-Ġ2sĠxĠ+Ġs
Rollout 36, Act: -21.417
=Ġ-864,ĠaddĠtoĠequationĠ(1'):ĊĊEquationĠ(d)Ġ+ĠequationĠ(
Rollout 24, Act: -21.794
,Ġa(r,Ġc)Ġâī¤Ġa(r,Ġc').ĊĊThen,ĠforĠ(R,ĠC)
Rollout 36, Act: -25.594
ĠtoĠfindĠrelationships.ĊĊFromĠ(1')Ġ-Ġ(2'):Ġ-2sĠxĠ+Ġ2sĠy

Layer 62

GATE_PROJ

Top 16 Positive Activations
Rollout 35, Act: 0.802
)ĠareĠrootsĠofĠ\(ĠQ(n)Ġ\).ĊĊGivenĠ\(ĠP(17)Ġ=Ġ10
Rollout 93, Act: 0.760
Ġ+Ġe^{-i\theta}}{2}\right)^{2014}Ġ=Ġ\frac
Rollout 35, Act: 0.709
24)Ġ=Ġ17Ġ\),ĠweĠfind:Ċ\[ĊQ(17)Ġ=Ġ
Rollout 90, Act: 0.563
and}Ġ\quadĠBĠ\equivĠ0Ġ\pmod{2018}ĊĠĠĠĠĠ\]Ċ
Rollout 17, Act: 0.519
ĠprobabilityĠdensityĠatĠtheĠoriginĠisĠ\(|\Psi(0)|^2Ġ=Ġ\frac{1}{\
Rollout 20, Act: 0.488
Ġ\(Ġp^{kn}Ġ\equivĠ1Ġ\pmod{47}Ġ\).ĠThisĠrequiresĠ\(
Rollout 19, Act: 0.426
frac{n(n+1)}{2}Ġ\right)^2Ġ\equivĠ17Ġ\pmod
Rollout 36, Act: 0.371
Ġ8064}{2s}Ġ\right)^2Ċ\]ĊĊAfterĠsimplificationĠandĠsolving
Rollout 59, Act: 0.369
Ġ=Ġ\left(\frac{1}{2}\right)^2Ġ\cdotĠ\frac{3}{7
Rollout 8, Act: 0.331
Ġ$5$ĠofĠtheĠcardsĠinĠaĠrowĠinĠaĠrandomĠorder.ĠSheĠwillĠbeĠhappyĠifĠandĠonly
Rollout 59, Act: 0.322
Ġ\left(\frac{4}{28}\right)^2Ġ\\Ċ&=Ġ\frac{9Ġ+
Rollout 4, Act: 0.312
^2}{(C_1Ġ+ĠC_2)^2}Ġ\quadĠ\text{and}
Rollout 4, Act: 0.288
C_1Ġ+ĠC_2}Ċ\]ĊĊGivenĠ\(ĠC_1Ġ=Ġ2000
Rollout 60, Act: 0.280
Hz}Ċ\]ĊĊTheĠbeatĠfrequencyĠisĠtheĠdifferenceĠbetweenĠtheĠreflectedĠfrequencyĠ\(Ġf''Ġ\)ĠandĠthe
Rollout 19, Act: 0.240
)(kĠ-Ġ4)}{2}Ġ\right)^2Ġ\equivĠ17Ġ\pmod
Rollout 5, Act: 0.213
ĠbĠ=Ġ175Ġ\).ĠSubstitutingĠtheseĠvalues:Ċ\[ĊxĠ=Ġ\sqrt{\
Top 16 Negative Activations
Rollout 70, Act: -7.922
2d1Ġ>=157Ġ=>d1Ġ>=78.5ĠButĠd1<=6,
Rollout 90, Act: -7.925
1,928=2018ÃĹ5000Ġ-8,072
Rollout 75, Act: -7.939
a2Ġ+b2Ġ+c2Ġ-3)/4.ĠButĠa,b,cĠareĠodd,Ġsquares
Rollout 75, Act: -7.953
2kĠ+1)2Ġ=4k2Ġ+4kĠ+1=4k(kĠ+1
Rollout 86, Act: -7.962
/9=7/9âīĪ0.777.ĊĊTherefore,ĠintegralĠcase7âīĪ0.
Rollout 45, Act: -7.976
ĠisĠf(x)=x3Ġ+4x2Ġ-4.ĠButĠcheckĠifĠwithĠthis,ĠdoĠthe
Rollout 82, Act: -7.986
8=80/38âīĪ2.105.ĠSoĠinĠthisĠcase,ĠtheĠbinary
Rollout 51, Act: -7.999
,ĠmaybeĠit'sĠpositiveĠ4/63Ġsin8θ?ĠLetĠmeĠcheck...ĠBecauseĠsometimesĠduring
Rollout 65, Act: -8.016
ĠtheĠnumberĠofĠdivisorsĠisĠ(1+1)^4=16.ĊĊButĠBut.ĠHowever,
Rollout 70, Act: -8.027
2kĠâī¡3Ġmod5,ĠsoĠkâī¡4Ġmod5Ġ(sinceĠ2*4=
Rollout 20, Act: -8.031
kn})=knĠ+1âī¡1ĠwhenĠkâī¡0Ġmod43.ĠButĠtheĠproblemĠrequiresĠfor
Rollout 36, Act: -8.039
764;Ġ=>xâīĪ2.0105...ĊĊButĠlikely,Ġsqrt(36
Rollout 65, Act: -8.045
ĠTheĠnumberĠofĠdivisorsĠisĠ(2+1)^4=8l.ĠAllĠtheĠdivisorsĠa
Rollout 75, Act: -8.065
ĠtheĠresultingĠ2k=Ġ(SĠ-3)/4=2*(a2Ġ+b2Ġ+
Rollout 68, Act: -8.137
Ġ(-4))x+Ġ4=Ġx2Ġ+8xĠ+4=0.ĠThen:ĊĊDisc
Rollout 45, Act: -8.154
m)x2Ġ-Ġ(5mĠ+1)xĠ-4Ġ-4mĠâī¥0ĊĊOr,ĊĊx

UP_PROJ

Top 16 Positive Activations
Rollout 94, Act: 1.312
c),ĠandĠ(d),ĠandĠtheĠanswerĠchoicesĠareĠcombinationsĠofĠthese.ĠFirst,ĠletĠmeĠrecallĠwhat
Rollout 42, Act: 1.296
ĠcyclopropaneĠring.ĠThatĠmightĠleadĠtoĠaĠconjugatedĠdieneĠorĠaĠlargerĠcarbocycle
Rollout 49, Act: 1.276
,ĠtheĠperfectĠsubsetĠpropertyĠholds?ĠInĠPolishĠspaces,ĠifĠit'sĠanĠuncountableĠPolishĠspace,
Rollout 1, Act: 1.273
:ĊĊ-2ãĢĪx.ĠThisĠactionĠwouldĠbeĠtheĠsameĠasĠaboveĠforĠlinearĠequations,ĠbutĠweĠalready
Rollout 42, Act: 1.272
Ġbond,ĠleadingĠtoĠaĠmoreĠconjugatedĠstructure.ĠTheĠmainĠproductĠwouldĠlikelyĠhaveĠtheĠmostĠstableĠalkene
Rollout 42, Act: 1.251
ĠcarbocationĠwouldĠcreateĠaĠlargerĠring.ĊĊAlternatively,ĠaĠhydrideĠshiftĠcouldĠoccurĠfromĠtheĠcyclo
Rollout 77, Act: 1.206
9).ĠSinceĠ266=14*19.ĠThereforeĠ14*19
Rollout 42, Act: 1.206
ĠspiroĠstructure.ĊĊAlternatively,Ġlet'sĠthinkĠofĠtheĠresult.ĠIfĠ(D)Ġspiro[3
Rollout 54, Act: 1.200
Ġbase.ĊĊAlso,ĠpropionateĠ(3)ĠisĠaĠveryĠweakĠbaseĠ(sinceĠpropionicĠacidĠisĠa
Rollout 26, Act: 1.199
ĠisĠtwo.ĠTheĠresultĠwouldĠbeĠ303.4,ĠwhichĠhasĠthreeĠsigĠfigs.
Rollout 25, Act: 1.180
67e-11ĊĊ=Ġ6.13e14Ġ/6.67
Rollout 24, Act: 1.169
leqslantĠa\left(r,Ġc^{\prime}\right)$ĠforĠallĠ$rĠ\
Rollout 97, Act: 1.162
^{-1.ĊĊPerĠtheĠIUPACĠdefinition,ĠreactionĠrateĠ(r)ĠisĠdefinedĠasĠrate=Ġrate
Rollout 86, Act: 1.137
pi/2)=pi/16âīĪ0.19635ĠâīĪ0.1
Rollout 54, Act: 1.133
3):ĊĊPropionateĠ(RCOOâĪĴ)ĠisĠtheĠweakestĠbase,ĠsoĠinĠpolarĠproticĠsolution,
Rollout 42, Act: 1.102
edĠandĠoneĠfour-memberedĠring.ĊĊAlternatively,ĠtheĠformationĠofĠaĠspiroĠcompoundĠmightĠnotĠevenĠrequire
Top 16 Negative Activations
Rollout 41, Act: -6.442
27)ĠseemsĠcorrect,ĠunlessĠthereĠisĠaĠmiscalculation.ĠThenĠIĠcanĠsearchĠtheĠcurrent
Rollout 79, Act: -6.474
Ġanagram.ĠSoĠ"MinorĠchanges"ĠcanĠbeĠrearranged.ĠLet'sĠtryĠtoĠsplitĠintoĠtwoĠparts
Rollout 14, Act: -6.488
Ġ\frac{f(x)}{\sqrt{xĠf(x)}}}$,ĠbutĠthatĠfeelsĠlikeĠaĠstretchĠand
Rollout 79, Act: -6.488
ĠfitsĠ8,4ĠisĠ"CHANGINGĠHANS".ĠHowever,ĠafterĠreconsideringĠandĠinĠfact
Rollout 78, Act: -6.518
22,ĠnĠ=Ġ27,ĠsoĠmĠ+ĠnĠ=Ġ49.ĠTheĠanswerĠis
Rollout 78, Act: -6.541
22ĠandĠnĠisĠ27,ĠsoĠmĠ+ĠnĠ=Ġ22Ġ+Ġ27
Rollout 89, Act: -6.554
.ĊĊTherefore,ĠunlessĠproblemĠisĠaltered,Ġthus:ĊĊSupposeĠthere'sĠmisprintĠinĠtheĠproblem.ĠInstead
Rollout 30, Act: -6.581
,ĠmaybeĠnot.ĠItĠdependsĠifĠthereĠisĠaĠmiscalculationĠhere.ĊĊLetĠmeĠcheckĠthroughĠeach
Rollout 34, Act: -6.603
Ġ+ĠC.ĊĊThus:ĊĊ1Ġ=Ġ2Ġ+ĠCĠ=>ĠCĠ=Ġ-1ĊĊTherefore,ĠtheĠsolution
Rollout 71, Act: -6.631
ĠThenĠ|a|Ġ+Ġ|b|Ġ+Ġ|c|.ĠĊĊGivenĠthat,ĠmaybeĠanotherĠapproach:Ġwith
Rollout 36, Act: -6.659
ĠperfectĠsquare,ĠbutĠourĠcalculationĠmightĠhaveĠbeenĠerror-prone.ĊĊAlternatively,ĠmaybeĠthereĠisĠanĠarithmeticĠmistake
Rollout 21, Act: -6.677
ĠanyĠrealĠnumberĠx,Ġthere'sĠanĠintegerĠkĠs.t.ĠxĠâĪĪĠ[k,Ġk+1
Rollout 67, Act: -6.718
Ġ3721d2.ĊĊSoĠsqrt(D)Ġ=Ġsqrt(3721d2
Rollout 79, Act: -6.759
Ġ(8?)ĠLetĠmeĠcheck:ĊĊĠINCIDENTĠâĨĴĠI-N-C-I-D-E-N-TĠâĢĵĠ8Ġletters
Rollout 79, Act: -6.937
4,ĠbutĠifĠ"Homing"ĠandĠ"Cranes"?ĠNo.ĠHmm.ĊĊAnotherĠapproach:
Rollout 79, Act: -7.040
Ġword.ĊĊAlternatively,Ġ"Machines"ĠandĠ"Rong"?ĠNotĠlikely.ĊĊAnotherĠangle:Ġ"Mech

DOWN_PROJ

Top 16 Positive Activations
Rollout 30, Act: 28.054
1\right)$ĠandĠ$b_{n+1}=a_{n}Ġb_{n-1
Rollout 16, Act: 24.727
,2)=15,Ġ(-1)^{3}=-1,ĠdenominatorĠ1Ġ-1/
Rollout 5, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 47, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 60, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 94, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 34, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 55, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 43, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 78, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 63, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 84, Act: 24.372
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 88, Act: 24.370
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 14, Act: 24.357
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 15, Act: 24.357
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Rollout 16, Act: 24.357
<|im_start|>systemĊYouĠareĠaĠhelpfulĠmathematicsĠassistant.Ċ<|im_start|>user
Top 16 Negative Activations
Rollout 52, Act: -14.430
2013ĠwithĠ4028ĠtermsĠsumĠtoĠresult.ĊĊButĠ-2013
Rollout 12, Act: -14.469
Ġ-Ġt)(tĠ+Ġa)(tĠ+Ġb)}Ġ}ĠĊĊButĠmaybeĠthat'sĠdiverging.ĊĊAlternatively
Rollout 30, Act: -14.490
Ġ}Ġ-1/a_{n+3}Ġ)ĊĊWeĠtriedĠexpressingĠeachĠterm,ĠandĠendedĠupĠfindingĠthat
Rollout 79, Act: -14.627
ĠpossibleĠletterĠcombinations.ĊĊH,ĠO,ĠM,ĠI,ĠN,ĠGĠ(homing).ĠThenĠremaining
Rollout 75, Act: -14.632
ĠintoĠtwoĠsquaresĠeach,ĠbutĠforĠ32ĠitĠcan'tĠbe,ĠbutĠinĠthatĠcase,Ġis
Rollout 75, Act: -14.671
Ġsquares.ĠThus,ĠforĠS=35,ĠitĠworks.ĠSimilarly,ĠforĠotherĠcases?ĊĊForĠexample
Rollout 71, Act: -14.862
3d2=2011,ĠgivenĠit'sĠprime,ĠtheĠonlyĠsolutionsĠareĠ(±4
Rollout 75, Act: -15.164
ĠeachĠ9ĠisĠaĠsingleĠsquare.ĠButĠfourĠsquares:ĊĊ16Ġ+9Ġ+9Ġ+1Ġ=
Rollout 39, Act: -15.454
9...9997101,ĠitĠcanĠbeĠsplitĠasĠ999...9
Rollout 11, Act: -15.458
Ġwork.ĠTryingĠaĠspecificĠexampleĠseemsĠnotĠpossibleĠasĠitĠresultedĠinĠprohibitedĠthings.ĠHmmm.ĊĊWait,
Rollout 65, Act: -15.634
1975,Ġc.ĊĊWhichĠhasĠratiostroubledĠasĠfollows:ĊĊĠoriginalĠprismĠ35
Rollout 95, Act: -15.652
ĠequalĠareas,ĠthusĠtheĠproblem'sĠconditionsĠareĠincorrect(butĠTheĠproblemĠsaysĠthatĠitĠcanĠbeĠsplitĠin
Rollout 66, Act: -17.928
ĠweĠendĠupĠwithĠbothĠlegsĠeven,ĠwhichĠconfilctsĠwithĠtheĠprimitiveĠtriple.ĠSoĠthisĠsuggestsĠthat
Rollout 93, Act: -17.948
)^kĠĠisĠzeroĠatĠx=x_k.ĠHmmm.ĊĊButĠx_kĠcorrespondsĠtoĠkĠfrom1
Rollout 49, Act: -23.696
ĠisĠatĠmostĠ2^{2^{d(X)}},ĠwhereĠd(X)ĠisĠtheĠdensity.ĠBut
Rollout 12, Act: -26.223
Ġ\nu^2)(1Ġ-Ġt^2)}}Ċ\]ĊĊNext,ĠweĠrecognizeĠthisĠintegralĠas

Layer 63

GATE_PROJ

Top 16 Positive Activations
Rollout 89, Act: 7.290
ĠresultĠinĠfractionsĠofĠpeopleĠwouldĠindicateĠaĠmiscalculation.ĠWhileĠthisĠproblemĠproducesĠ66.(
Rollout 45, Act: 7.133
Ġ+ĠbĠ+Ġ2cĠ+Ġ4ĊĊSimplify:Ġ20Ġ=Ġ6Ġ+Ġ
Rollout 25, Act: 7.109
).ĠSoĠinĠrealĠcase,ĠtheĠtrajectoryĠisĠhyperbolaĠwithĠperigeeĠr_pĠandĠtheĠap
Rollout 52, Act: 7.095
=Ġ-33.5.ĠNotĠinteger.ĠDisregard.ĊĊ9.Ġk=212
Rollout 68, Act: 7.062
2Ġ+Ġ4xĠ+Ġ4.ĠRearrangingĠterms,ĠweĠget:ĊĊx^2Ġ+
Rollout 14, Act: 7.052
ĠbitĠtricky.ĠMaybeĠexpressĠthisĠas:ĊĊRearranged,ĊĊ$$Ċf(x)Ġ=ĠaĠ-
Rollout 19, Act: 7.000
-5=Ġ-4ĠwhichĠisnâĢĻtĠpositive.ĠDisregard.Ġk=83,ĠwhichĠgives
Rollout 88, Act: 6.997
licativeĠparameters,ĠbiasesĠareĠadditive.ĠSoĠmaybeĠtheĠquestionĠisĠonlyĠaskingĠaboutĠtheĠweights.ĠThenĠitĠwould
Rollout 8, Act: 6.995
ĠbeĠadjacent).ĠButĠtheĠwayĠtheĠproblemĠisĠphrased:Ġ"allĠtheĠredĠcardsĠlaidĠoutĠare
Rollout 89, Act: 6.946
Ġ-Ġ(2/5)*T,Ġright?ĊĊSimplifyingĠthat,ĠboysĠ=Ġ(1Ġ-Ġ
Rollout 89, Act: 6.909
ĠthatĠothersĠmayĠgot.ĊĊAlternatively,ĠperhapsĠmyĠovercomplicating.ĊĊWait,ĠisĠitĠpossibleĠinĠnon
Rollout 25, Act: 6.872
2/2,ĠbutĠsinceĠit'sĠaĠhyperbolicĠtrajectory,ĠεĠisĠpositive.ĊĊThen,Ġwe
Rollout 42, Act: 6.844
ĠveryĠunstable.ĠSoĠperhapsĠitĠundergoesĠaĠhydrideĠshift?ĠLet'sĠsee.ĊĊWait,Ġwhen
Rollout 70, Act: 6.821
1*7Ġ+Ġd_0).ĊĊRearranging,ĊĊd_k*(10^kĠ-
Rollout 44, Act: 6.805
ĠtheĠirreversibleĠprocess.ĠWait,ĠbutĠthatĠseemsĠcounterintuitive,ĠbecauseĠinĠirreversibleĠprocessesĠthereĠareĠlosses.
Rollout 81, Act: 6.789
ĠBADĠandĠCADĠareĠacute,ĠsoĠsumĠcanĠbeĠacuteĠorĠpossiblyĠgreaterĠthanĠ90,ĠbutĠgiven
Top 16 Negative Activations
Rollout 46, Act: -0.471
ĠgetsĠcomplicated.ĊĊAlternatively,ĠtheĠquestionĠisĠaskingĠforĠtheĠLaplacianĠofĠU,ĠbutĠLaplace's
Rollout 59, Act: -0.486
ĠPhilĠhaveĠtwoĠfairĠcoinsĠandĠaĠthirdĠcoinĠthatĠcomesĠupĠheadsĠwithĠprobabilityĠ$\frac47$Ġ.
Rollout 87, Act: -0.518
AfterĠfirstĠintegratingĠwithĠrespectĠtoĠx:ĊĊTheĠintegralĠâĪ«0^aĠdxĠ/Ġsqrt(x2Ġ+
Rollout 46, Act: -0.518
L/âĪĤxĠ=ĠÏīĠv_yĠ+ĠâĪĤU/âĪĤxĊĊTherefore,Ġequation:ĊĊ
Rollout 49, Act: -0.526
ĠX.ĠĠAĠĠHausdorffĠspaceĠwithĠGδĠdiagonalĠisĠfirst-countable.ĠAh,
Rollout 57, Act: -0.548
ĠofĠitsĠpoints.ĊĊHowever,theĠCantorĠsetĠCĠhasĠmeasureĠzero.ĠSo,everyĠneighbourhoodĠofĠaĠpoint
Rollout 46, Act: -0.565
/âĪĤxĠ=ĠÏīĠdy/dtĠ-ĠâĪĤU/âĪĤxĊĊTherefore,ĠputtingĠinto
Rollout 3, Act: -0.566
Ġ\)ĠisĠtheĠmolarĠmassĠofĠradonĠinĠkg/mol.ĊĊ**Steps:**ĊĊ1.Ġ**
Rollout 60, Act: -0.568
Ġ\mathrm{~ms}^{-1}$.ĠTheĠreflectedĠsignalĠisĠreceivedĠbyĠtheĠsourceĠandĠsuperposed
Rollout 49, Act: -0.608
ĠÏĥ-compact.ĊĊHence,ĠXĠisĠéĽ-dimensional?ĠMaybeĠnotĠnecessarily.ĊĊAlternatively,Ġembed
Rollout 80, Act: -0.628
Ġsurface.ĊĊInĠtheĠmovieĠ2012,ĠneutrinosĠhaveĠmutatedĠandĠnowĠareĠcompletelyĠabsorbedĠby
Rollout 47, Act: -0.646
ĠsidesĠandĠtheĠwallsĠofĠtheĠtunnel.ĠAlso,ĠtheĠairĠflowĠwithĠrespectĠtoĠtheĠtrainĠisĠsteadyĠand
Rollout 97, Act: -0.661
Ġatm.ĊĊUnderĠstoiechmentricĠdecomposition,ĠyĠisĠtheĠextentĠofĠreactionĠinĠtermsĠofĠhowĠmany
Rollout 97, Act: -0.702
Ġif.ĊĊAssumption:ĠifĠN2O5ĠĠfollowsĠP(t}=Ġexp(-kt}.,Ġthen
Rollout 44, Act: -0.900
1}$ĠtoĠ$\mathrm{V}_{2}$ĠunderĠadiabaticĠconditionsĠasĠcomparedĠtoĠthatĠwhen
Rollout 13, Act: -0.987
ĠisĠ$12\%$.ĠĠSheĠmakesĠnoĠpaymentsĠforĠ4Ġyears,ĠbutĠhasĠtoĠpay

UP_PROJ

Top 16 Positive Activations
Rollout 40, Act: 4.885
,ĠtheĠtermĠwasĠ10/(1Ġ-ĠÏī)*(1Ġ+Ġi).ĠWait,Ġearlier:ĊĊ
Rollout 34, Act: 4.798
ĠâĪ«Ġe^{2s}ĠdsĠfromĠ0ĠtoĠtĠisĠ(1/2)(e^{
Rollout 40, Act: 4.783
1Ġ)ĊĊHence,Ġ1/(1Ġ-ĠÏī)Ġ=Ġ[1Ġ+Ġi(âĪļ2
Rollout 40, Act: 4.773
^nĠz0Ġ+ĠbĠ(sum_{k=0}^{n-1}Ġa^kĠ).
Rollout 8, Act: 4.716
Ġk)Ġ=Ġ5!Ġ/Ġ(5Ġ-Ġk)!.ĊĊSimilarly,ĠarrangeĠ(5Ġ-k)
Rollout 40, Act: 4.595
ļ2ĊĊTherefore,Ġ1/(1Ġ-ĠÏī)Ġ=Ġ[Ġ(1Ġ-ĠâĪļ2
Rollout 23, Act: 4.524
147Ġ/Ġ0.098ĠâīĪĠ7.073...ĠminâĪĴ
Rollout 69, Act: 4.514
38Ġ/Ġ0.9848ĠâīĪĠ0.00343.ĊĊ
Rollout 8, Act: 4.482
(n,k)Ġ=Ġn!Ġ/Ġ(nĠ-Ġk)!.ĊĊSimilarly,ĠforĠaĠdifferentĠcolorĠsequenceĠwith
Rollout 92, Act: 4.465
iĠ=Ġ200/-123ĠâīĪĠ-1.6260...ĊĊThus
Rollout 89, Act: 4.453
Ġ2/5Ġ*500/3ĠâīĪ200/3âīĪ66.
Rollout 40, Act: 4.397
ThenĠpluggedĠintoĠtermĠ[10/(1Ġ-ĠÏīĠ)]*(1Ġ+i).ĠThenĠmultiplyingĠout
Rollout 58, Act: 4.390
ĠtotalĠnumberĠofĠquestionsĠtoĠbeĠtheĠsumĠfromĠ1ĠtoĠ2018.ĠButĠjustĠto
Rollout 13, Act: 4.385
0*(1Ġ+Ġ0.12/4)^(4*4)ĊĊFirst,ĠcalculateĠ
Rollout 44, Act: 4.380
γ-1)=ĠT2*(20/1)^(2/3).ĠButĠT2=
Rollout 92, Act: 4.371
Ġ1/(-0.615)ĠâīĪ5.405Ġ-1.6
Top 16 Negative Activations
Rollout 10, Act: -2.884
bbles.ĠTheĠkeyĠisĠtoĠidentifyĠlosingĠpositionsĠ(P-positions)ĠwhereĠeveryĠmoveĠleadsĠtoĠa
Rollout 87, Act: -2.887
ĠconstantĠinvolvingĠlogarithmicĠterms.ĠAfterĠevaluatingĠthisĠintegralĠ(aĠknownĠresultĠinĠelectromagnetism),ĠtheĠpotential
Rollout 98, Act: -2.888
**ĠĠĊĠĠĠ-ĠAĠwhiteĠdwarfĠprogenitorĠwithĠ**0.85Msun**ĠisĠinconsistent:
Rollout 42, Act: -2.894
ocationĠrearrangement**Ġoccurs.ĠThisĠrearrangementĠinvolvesĠ**ring-openingĠofĠtheĠcyclopropane**Ġto
Rollout 82, Act: -2.895
Ġ-ĠEachĠgroupĠisĠencodedĠwithĠaĠbinaryĠprefixĠcodeĠ(0ĠorĠ1),ĠresultingĠinĠtheĠfollowingĠcod
Rollout 3, Act: -2.901
oundingĠtoĠSignificantĠFigures:**ĊĠĠĠGivenĠtheĠinputsĠ(molarĠmassĠ\(222Ġ\,Ġ\
Rollout 22, Act: -2.966
ĠgasĠ(H2)ĠatĠstandardĠtemperatureĠandĠpressureĠ(STP),ĠweĠuseĠtheĠformulaĠforĠtheĠmean
Rollout 48, Act: -2.987
Ġ**TotalĠLength:**Ġ`0034`Ġ(52ĠbytesĠtotalĠpacketĠlength).ĊĠĠĠ-Ġ**
Rollout 6, Act: -2.998
.ĊĊAnotherĠangle:ĠLetĠS_LĠcontainĠtheĠtupleĠ<n,Ġα_nĠ>.ĠButĠencodeĠthisĠin
Rollout 10, Act: -3.014
positions)ĠwhereĠeveryĠmoveĠleadsĠtoĠaĠwinningĠpositionĠ(N-position)ĠforĠtheĠopponent.ĊĊ1.Ġ**
Rollout 98, Act: -3.033
un)ĠwithĠlongĠlifetimes.ĠTheirĠlifetimesĠ(10ĠGyr,Ġ~24ĠGyr
Rollout 22, Act: -3.080
-Ġ\(ĠTĠ\)ĠisĠtheĠtemperatureĠinĠKelvinĠ(assumedĠtoĠbeĠ273.1
Rollout 10, Act: -3.151
**:ĊĠĠĠ-ĠPlayersĠtakeĠturnsĠremovingĠsquareĠnumbersĠ(1,Ġ4,Ġ9,Ġ1
Rollout 65, Act: -3.156
995Ġitself.ĠHalfĠofĠtheseĠdivisorsĠ(excludingĠtheĠmidpoint)ĠareĠlessĠthanĠ19
Rollout 6, Act: -3.485
ĠoracleĠcanĠcontainĠthoseĠpositions.ĠSo,ĠforĠeachĠ(n,Ġi),ĠwhereĠiĠisĠaĠbitĠposition
Rollout 48, Act: -3.543
4Ġ(IPv4),ĠIHLĠisĠ5Ġ(20Ġbytes).ĊĠĠĠ-Ġ**TypeĠofĠService

DOWN_PROJ

Top 16 Positive Activations
Rollout 79, Act: 33.036
Ġthinking,ĠtheĠintendedĠanswerĠisĠ**MONITOR'SĠGAZE**,ĠbutĠtheĠdefinitiveĠanswer,ĠfollowingĠstandardĠcrossword
Rollout 79, Act: 30.549
.ĊĊAnotherĠidea:ĠTheĠsolutionĠisĠ"MACHINEĠGUNS".Ġ"Machine"Ġ(7Ġletters)Ġand
Rollout 95, Act: 25.314
0为1,æ¯ı个ĠtrapezoidalĠregionĠéľĢè¦ģs1,Ġs2çŃīãĢĤĊĊ
Rollout 14, Act: 23.731
Ġg(y).ĠThenĠJĠscalesĠwithĠd^2ĠâĪ«g^2Ġdx=Ġd^2/c
Rollout 79, Act: 23.450
ĠS.ĠThenĠ4-letterĠwordĠcouldĠbeĠ"gros"ĠâĢĶĠno.ĠNotĠplausible.ĊĊAlternatively,
Rollout 9, Act: 23.330
$Ġunits.ĠĠIsoscelesĠtriangleĠ$GEM$ĠhasĠbaseĠ$EM$Ġ,ĠandĠthe
Rollout 81, Act: 22.441
ãĢĤĠButĠasĠangleĠBACĠisĠrequiredĠtoĠbeĠθâīĪ72Ġdegrees.ĊĊĠwhichĠcontradictsĠbecause
Rollout 49, Act: 22.364
}ĠatĠx.ĠForĠeachĠn,ĠU_nĠâĪ©ĠAĠisĠuncountable.ĠLetâĢĻsĠchoose
Rollout 44, Act: 21.677
ĠadiabaticĠconditionsĠwithĠ$\mathrm{T}_{1}Ġ\neqĠ\mathrm{T}_{2}
Rollout 79, Act: 21.429
Ġdon'tĠhave.ĠĊĊScraps:Ġ"MorningĠGCA"ĠâĢĶĠnothing.ĠĊĊORĠsomeĠtermsĠlike
Rollout 82, Act: 21.426
ä½Ĩéľįå¤«æĽ¼ç®Ĺæ³ķéĢļ常ä¼ļåĢ¾åIJij让è¾ĥé«Ļæ¦Ĥçİĩ符åı·çĶ¨è¾ĥçŁŃçļĦ代çłģãĢĤä½Ĩæĺ¯åľ¨
Rollout 57, Act: 20.934
xå¤Ħ为1,并ä¸ĶAâĬĨP?ĊĊĠçĦèĢĮ,åĽłä¸ºæĹłçIJĨæķ°çļĦéĽĨåIJĪ
Rollout 56, Act: 20.563
ĠaĠgeneralĠformula.ĊĊFirstĠtriangle:ĠS0Ġ=ĠâĪļ3Ġ/2Ġ~0.866
Rollout 14, Act: 20.207
Ġ(âĪ«Ġf(x)ĠdxĠ)^3Ġ=ĠâĪ«Ġf(x)ĠdxĠ*ĠâĪ«Ġf(y
Rollout 14, Act: 20.193
ĠâĪ«0^âĪŀĠf(x)ĠdxĠ=ĠâĪ«0^âĪŀĠf(x)Ġ\cdot1
Rollout 33, Act: 20.058
Ġpermutation.ĠWhichĠmatchesĠf(2)=1.ĊĊForn=3ĠasĠearlier,it's3.ĊĊLet
Top 16 Negative Activations
Rollout 60, Act: -20.011
ĠtheĠDopplerĠeffectĠformulaĠforĠaĠmovingĠsource:Ċ\[Ċf''Ġ=Ġf'Ġ\frac{v
Rollout 48, Act: -20.267
4Ċ3.ĠB612Ġ=Ġ0xB612Ċ4.Ġ40
Rollout 55, Act: -20.353
}\),ĠresultingĠin:ĊĠĠ\[ĊĠĠ\begin{pmatrix}Ġ\frac{1}{2
Rollout 79, Act: -21.079
-letterĠwordsĠfromĠtheseĠlettersĠinclude:ĊĊ-Ġ"Hernias"ĠplusĠsomeĠothers.ĠNo.ĠMaybeĠ"
Rollout 26, Act: -21.338
pyĠchangeĠwhenĠgoingĠfromĠHg(l)ĠtoĠHg(g),ĠwhichĠisĠtheĠenthalpyĠof
Rollout 48, Act: -21.642
63,814ĠinĠhexĠisĠ0xF946.ĠWait,ĠthisĠisĠa
Rollout 48, Act: -21.699
Ġ0x4000.ĠSoĠ0xFB46Ġ+0x4000
Rollout 32, Act: -21.747
Ïī.ĊĊSoĠM2Ġis:ĊĊ\[Ċ\begin{pmatrix}Ġ0Ġ&Ġ1Ġ\\
Rollout 48, Act: -21.918
0ĠtoĠ0xFFFF?),ĠwaitĠno,Ġ0xFB46ĠisĠaĠ16-bitĠvalue
Rollout 55, Act: -22.103
Ġ\frac{1}{2}Ġ\end{pmatrix}Ċ\]ĊĊYep,ĠsoĠthatĠlooksĠabout
Rollout 49, Act: -22.666
endixsonĠderivative.ĊĊTheĠCantor-BendixsonĠtheoremĠstatesĠthatĠinĠaĠseparableĠmetr
Rollout 11, Act: -24.030
}Ġ=Ġ1/(\overline{ab})).ĠTherefore,ĠmaybeĠexpressionsĠcanĠbeĠwrittenĠinĠterms
Rollout 32, Act: -24.282
omegaĠ\\Ġ1Ġ&Ġ0Ġ\end{pmatrix}.\]FindĠtheĠsumĠofĠtheĠentries
Rollout 48, Act: -24.371
0x4000:ĠsumĠbecomesĠ0xFB46Ġ+Ġ0x400
Rollout 17, Act: -26.246
ĠmagnitudeĠofĠtheĠintegralĠ\(|\mathbf{I}|\)ĠisĠ\(\boxed{2.5
Rollout 17, Act: -26.879
ĠmagnitudeĠofĠtheĠintegralĠ\(|\mathbf{I}|\)ĠforĠtheĠgivenĠproblem,ĠweĠneedĠto